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TRANSCRIPT
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Measuring mobility
Austin Nichols
July 31 2014
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Topics
I transition matrices
I matrix-based mobility measures
I other mobility measures
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Focus
We will examine various means of measuring mobility with a focus oneconomic mobility of individuals over time primarily due to changes in income
But most of these measures can be applied in many areas and are used forexample to measure changes in measured teacher quality in educationresearch or any changing states such as marital status or occupation disabilityor morbidity or changing prices or market shares
Many (but not all) of commonly used measures rely on an estimated transitionmatrix so letrsquos start there
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition matrices
The usual setup for a transition matrix is to measure status s at time t minus 1 andagain at time t then to estimate the matrix M1 (the one denotes a one-perioddelta)
st = M1stminus1
in which case each column of M1 sums to one or sometimes its transpose
sTt = sTtminus1MT1
in which case each row of MT1 sums to one
s may measure for example which fifth (or half or hundredth) of the incomedistribution a panel survey respondent falls into in one year and then whichfifth they fall in the next year
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Alternative transition matrices
One could also measure which fifth of the income distribution a respondentfalls into then which fifth their child appears in 30 years later(intergenerational mobility)
Or we could measure which fifth of a different economic status distribution arespondent falls into eg a measure of educational attainment then whichfifth of the income distribution their child appears in 30 years later (in whichcase we do not have s on both sides of that equation)
Or states defined by an absolute measure such as the US poverty line (unlikethe traditional European model for measuring poverty the US cutoff is definedby a theoretical lower bound budget adjusted only for measured inflation) Thiscan define multiple states as well we can look at who is poornonpoor in eachperiod but we can also look at below the poverty line [12) times the povertyline [23) times the poverty line etc
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One (related) alternative to transition matrices
We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix
Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Topics
I transition matrices
I matrix-based mobility measures
I other mobility measures
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Focus
We will examine various means of measuring mobility with a focus oneconomic mobility of individuals over time primarily due to changes in income
But most of these measures can be applied in many areas and are used forexample to measure changes in measured teacher quality in educationresearch or any changing states such as marital status or occupation disabilityor morbidity or changing prices or market shares
Many (but not all) of commonly used measures rely on an estimated transitionmatrix so letrsquos start there
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition matrices
The usual setup for a transition matrix is to measure status s at time t minus 1 andagain at time t then to estimate the matrix M1 (the one denotes a one-perioddelta)
st = M1stminus1
in which case each column of M1 sums to one or sometimes its transpose
sTt = sTtminus1MT1
in which case each row of MT1 sums to one
s may measure for example which fifth (or half or hundredth) of the incomedistribution a panel survey respondent falls into in one year and then whichfifth they fall in the next year
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Alternative transition matrices
One could also measure which fifth of the income distribution a respondentfalls into then which fifth their child appears in 30 years later(intergenerational mobility)
Or we could measure which fifth of a different economic status distribution arespondent falls into eg a measure of educational attainment then whichfifth of the income distribution their child appears in 30 years later (in whichcase we do not have s on both sides of that equation)
Or states defined by an absolute measure such as the US poverty line (unlikethe traditional European model for measuring poverty the US cutoff is definedby a theoretical lower bound budget adjusted only for measured inflation) Thiscan define multiple states as well we can look at who is poornonpoor in eachperiod but we can also look at below the poverty line [12) times the povertyline [23) times the poverty line etc
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One (related) alternative to transition matrices
We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix
Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Focus
We will examine various means of measuring mobility with a focus oneconomic mobility of individuals over time primarily due to changes in income
But most of these measures can be applied in many areas and are used forexample to measure changes in measured teacher quality in educationresearch or any changing states such as marital status or occupation disabilityor morbidity or changing prices or market shares
Many (but not all) of commonly used measures rely on an estimated transitionmatrix so letrsquos start there
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition matrices
The usual setup for a transition matrix is to measure status s at time t minus 1 andagain at time t then to estimate the matrix M1 (the one denotes a one-perioddelta)
st = M1stminus1
in which case each column of M1 sums to one or sometimes its transpose
sTt = sTtminus1MT1
in which case each row of MT1 sums to one
s may measure for example which fifth (or half or hundredth) of the incomedistribution a panel survey respondent falls into in one year and then whichfifth they fall in the next year
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Alternative transition matrices
One could also measure which fifth of the income distribution a respondentfalls into then which fifth their child appears in 30 years later(intergenerational mobility)
Or we could measure which fifth of a different economic status distribution arespondent falls into eg a measure of educational attainment then whichfifth of the income distribution their child appears in 30 years later (in whichcase we do not have s on both sides of that equation)
Or states defined by an absolute measure such as the US poverty line (unlikethe traditional European model for measuring poverty the US cutoff is definedby a theoretical lower bound budget adjusted only for measured inflation) Thiscan define multiple states as well we can look at who is poornonpoor in eachperiod but we can also look at below the poverty line [12) times the povertyline [23) times the poverty line etc
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One (related) alternative to transition matrices
We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix
Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition matrices
The usual setup for a transition matrix is to measure status s at time t minus 1 andagain at time t then to estimate the matrix M1 (the one denotes a one-perioddelta)
st = M1stminus1
in which case each column of M1 sums to one or sometimes its transpose
sTt = sTtminus1MT1
in which case each row of MT1 sums to one
s may measure for example which fifth (or half or hundredth) of the incomedistribution a panel survey respondent falls into in one year and then whichfifth they fall in the next year
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Alternative transition matrices
One could also measure which fifth of the income distribution a respondentfalls into then which fifth their child appears in 30 years later(intergenerational mobility)
Or we could measure which fifth of a different economic status distribution arespondent falls into eg a measure of educational attainment then whichfifth of the income distribution their child appears in 30 years later (in whichcase we do not have s on both sides of that equation)
Or states defined by an absolute measure such as the US poverty line (unlikethe traditional European model for measuring poverty the US cutoff is definedby a theoretical lower bound budget adjusted only for measured inflation) Thiscan define multiple states as well we can look at who is poornonpoor in eachperiod but we can also look at below the poverty line [12) times the povertyline [23) times the poverty line etc
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One (related) alternative to transition matrices
We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix
Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Alternative transition matrices
One could also measure which fifth of the income distribution a respondentfalls into then which fifth their child appears in 30 years later(intergenerational mobility)
Or we could measure which fifth of a different economic status distribution arespondent falls into eg a measure of educational attainment then whichfifth of the income distribution their child appears in 30 years later (in whichcase we do not have s on both sides of that equation)
Or states defined by an absolute measure such as the US poverty line (unlikethe traditional European model for measuring poverty the US cutoff is definedby a theoretical lower bound budget adjusted only for measured inflation) Thiscan define multiple states as well we can look at who is poornonpoor in eachperiod but we can also look at below the poverty line [12) times the povertyline [23) times the poverty line etc
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One (related) alternative to transition matrices
We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix
Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One (related) alternative to transition matrices
We could measure at which point in the overall distribution at time t + 1 (orgeneration t + 1) each conditional quantile reaches calculating quantilesconditional on starting points this is similar in spirit to a transition matrix
Can do this nonparametrically with a series of kernel-weighted quantileregressions as in Nichols and Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Nichols and Favreault (2009)
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1935-39
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1940-44
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1945-49
02
46
81
0 20 40 60 80 100
Rank Sum Parents Years of Education
Avg earnings position in birth cohort1950-54
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
One alternative to transition matrices
That paper also critiques the common ldquointergenerational elasticityrdquo measurewhich regresses log income of the child on log income of the parent (note thatwe could do this for income of a person at two points in time as well tomeasure ldquointragenerational elasticityrdquo)
Making sure the nonparametric quantile regressions satisfy some basicadding-up constraints is no easy matter and is not even attempted in Nicholsand Favreault (2009)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quintile transition matrices
The canonical example defines st so it measures fifths of an incomedistribution so M1 in
st = M1stminus1
is a quintile transition matrix (all the same theory applies to any quantiletransition matrix but 5 categories seems to be the optimal number for ourlimited attention) In that case s = (02 02 02 02 02)T in every period andM1 must be bistochastic ie rows and columns must sum to one
Now there is no information in s ie the ldquomiddle classrdquo cannot grow or shrinkif it is always 60 percent of the population and we can devote our attentionexclusively to the properties of M1
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying transition matrices
We donrsquot need to show the numbers in a table of course transition bar chartscommon in reports
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Displaying comparisons of transition matrices
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov matrices
Much of the appeal of transition matrices arises from the idea that we can usethem to describe longer-run dynamics In particular if the probability a personwinds up in a particular row of st depends only on which row of stminus1 they arein then M1 is a Markov matrix and we can describe the probability they windup in a particular row at time t + 1
st+1 = M1st = M1M1stminus1 = M21 stminus1
or time t + k
st+k = Mk1 st
so we donrsquot need to compute two-period transition matrix M2 or thethree-period transition matrix M3 or measure any longer transitions
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
Unfortunately it rarely is the case that the data satisfies the Markov assumptionthat history does not matter for transition rates even at the first comparisonpoint of the two-period transition matrix M2 (recall the subscript describes thedelta in time periods whereas the superscript denotes the power ie M2 is theproduct of M1 and M1 the square of single-period transition matrices)
st+1 = M2stminus1 6= M21 stminus1
but sometimes it is possible to expand the space over which the states in s aremeasured and get a transition matrix that comes close to satisfying the Markovassumption
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Markov failures
For example instead of measuring poornonpoor in one period and trying topredict poornonpoor in the next period and all future periods we can measurepoornonpoor in two periods and try to predict the next two periods and allfuture periods (instead of two proportions in s now there are 4)
The meaning of the matrix changes in that case of course but it allowslong-run projections if Markov assumptions are satisfied
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Transition estimation
Stata has a command xttrans for measuring transition rates but it does notrespect the panel structure properly tabulate works fine though and svytabulate allows weights cluster-robust standard errors and tests of hypotheses
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
gen above=ln_wagegtm if ln_wagelt
gen lastyr=labove
bysort idcode (year) gen wrong=above[_n-1]
gen nextyr=fabove
tab above nextyr nofreq row
tab lastyr above nofreq row
tab wrong above nofreq row
xttrans above
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
xttrans
tab lastyr above nofreq row
| above
lastyr | 0 1 | Total
-----------+----------------------+----------
0 | 8004 1996 | 10000
1 | 1285 8715 | 10000
-----------+----------------------+----------
Total | 4479 5521 | 10000
tab wrong above nofreq row
| above
wrong | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
xttrans above
| above
above | 0 1 | Total
-----------+----------------------+----------
0 | 7833 2167 | 10000
1 | 1714 8286 | 10000
-----------+----------------------+----------
Total | 4741 5259 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
Of course tabulate does not know that the resulting matrix is supposed to bebistochastic in the previous example and so it predicts 45 percent will be belowthe median next period and 55 percent above which cannot happen
This error can be due to a mass of people right at quantile breaks (if a lot ofpeople are right at the median wage) and tie-breaking rules or an unbalancedpanel
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
The latter problem is pervasive and requires some thought how do you wantto select a balanced panel to do your estimation (a larger issue in general)
One simple and defensible method is to reweight the data using the proportionof people in each category in the first time period who also appear in thesecond (dropping those who appear only in one period) This is anonparametric propensity score approach to nonresponseattrition adjustment(see also Nichols 2007) and can work to create a representative balanced panelof any length (representing the population of the base year) Properly done italso requires re-estimating all the relevant quantiles for each set of balanceddata using weights
A quick and dirty approach is to simply estimate the quantiles for thesubsample in period t + 1 that has data available in period t which turns outto fix most problems
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment
webuse nlswork clear
keep if inlist(year707172737778)
egen m=median(ln_wage) by(year)
egen m2=median(ln_wage) if lln_wagelt by(year)
gen above=ln_wagegtm if ln_wagelt
gen above2=ln_wagegtm2 if ln_wagelt
gen nextyr=fabove
gen nexty2=fabove2
tab above nextyr if year==70 nofreq row
tab above nexty2 if year==70 nofreq row
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
A quick and dirty adjustment cont
tab above nextyr if year==70 nofreq row
| nextyr
above | 0 1 | Total
-----------+----------------------+----------
0 | 7394 2606 | 10000
1 | 1341 8659 | 10000
-----------+----------------------+----------
Total | 4167 5833 | 10000
tab above nexty2 if year==70 nofreq row
| nexty2
above | 0 1 | Total
-----------+----------------------+----------
0 | 8192 1808 | 10000
1 | 2240 7760 | 10000
-----------+----------------------+----------
Total | 5019 4981 | 10000
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesAlternativesQuantile transition matricesMarkov matricesEstimation
Quantile transition matrix estimation
One can also use optimize to estimate the closest matrix (where close is definedusing the spectral norm measure of distance for matrices) to the empiricalestimate that is bistochastic but these small deviations are unlikely to matter inpractice (at least I have never seen any real difference by moving to a properlyconstrained estimate of the quantile transition matrix in my own work)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Matrix properties
Wersquove estimated a matrix now what Is the society highly mobile or not
Perfect immobility would be an identity matrix as the transition matrix perfectmobility might be any matrix with zeros on the diagonal (no one ends wherethey started) or everyone has equal probability of winding up in the variouspossible slots next period regardless of starting positions
Any other matrix has a large-dimensional set of possible deviations from theseideals Hard to look at a pair of transition matrices and say ldquothis matrixcorresponds to an unambiguously more mobile society than that onerdquo
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility statistics for transition matrices
Shorrocks (1978 Econometrica) proposed measures of mobility based onquantile transition matrices which generated a literature on matrix-basedmobility measures notably including work on ordering due to Dardanoni(1993) with a social welfare foundation
Sommers and Conlisk (1979) and Bartholomew (1982) also defined mobilitymeasures based on a quantile transition matrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Transition matricesMobility measures
Mobility measures
Letrsquos denote each of the commonly used measures with a single letter
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
All are easy to calculate in Mata or Stata given the estimate of the transitionmatrix
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Other mobility measures
Shorrocks (1978 Journal of Economic Theory) defined mobility in terms ofreductions in an inequality measure due to changes in accounting period Thisdefinition of mobility or a related one from Maasoumi and Zandvakili (1986) isused in many articles eg Burkhauser and Poupore (1997) Maasoumi andTrede (2001) and Kopczuk Saez and Song (2010) Call this the ldquoratiordquomeasure R
Nichols (2008 2010) defined mobility risk in terms of variability of growthpaths denote this M
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008 2010) measure
The goal of my own approach was to find a measure for income mobility thatwould integrate measures of mobility rates volatility and long-run inequality
The central insight was that an inequality measure that is additivelydecomposable by population subgroup such as the generalized entropy indexwith parameter 2 (GE2) or half the squared coefficient of variation can beapplied to panel data
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols measure cont
Let individuals be the subgroups and then the ldquobetween-grouprdquo inequalitycomponent measures long-run inequality across people while theldquowithin-grouprdquo inequality component measures inequality in individual incomeover time a combination of mobility and volatility
If we measure mobility risk as the variance of growth rates divided by squaredmean income the GE2 decomposition maps perfectly onto a regressionframework
yit = ui + ri t + eit
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Nichols (2008) graphic
Variation inGrowth Rates Variation in
Mean Incomes
Variation around trend0
1020
3040
5060
7080
Inco
me
-4 -3 -2 -1 0 1 2 3 4 5
TimeAustin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
This measure needs panel data of course but with a long panel we canmeasure the components of income risk using short windows say 3 or 5 yearsand compute at each overlapping window of time for estimates of theevolution of inequality volatility and mobility risk over time
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
05
1
15
2
25
3
R
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
R=Aggregate risk
0
05
1
15
2
25
3
I
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
I=Long-run inequality
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing across time and space
0
01
02
03
V
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
V=Variability around trend
0
01
02
03
M
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Year
United States CanadaGreat Britain Germany
M=Mobility risk
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Mobility measures
Adding to our list
I T Trace measure [m minus Tr(M)](m minus 1) from Shorrocks (1978 Ecm)
I D Determinant measure det(M)(m minus 1) from Shorrocks (1978 Ecm)
I E Eigenvalue measure one minus the modulus of the second largesteigenvalue of M due to Sommers and Conlisk (1979)
I M Mean crossing measure the sum over i and j (from 1 to m) of Mij
times |i minus j | divided by m(m minus 1) due to Bartholomew (1982)
I R Ratio of multi-period to weighted average single-period inequality(Shorrocks 1978 JET)
I M Mobility risk (Nichols 2008 2010 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures
All of these measure different concepts of mobility and will rate mobility in thesame data differently
For example if we ask which country has the highest level of economic mobilityin recent data we can get quite different rankings using different measures
That said they are all highly correlated in actual empirical examples (Nichols2008 Nichols and Rehm 2014)
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Comparing measures for 30 countries
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SwedenDenmark
FinlandPortugal
LuxembourgNorway
GermanyCanada
BelgiumGreat BritainNetherlandsSwitzerlandCzech Rep
IcelandHungary
AustriaAustralia
CyprusItaly
FranceIreland
SpainGreece
SlovakiaEstoniaPoland
United StatesLithuania
LatviaKorea
-3 -2 -1 0 1 2 3Standardized measure
M=Mobility risk T=Trace
D=Determinant R=Ratio
E=Eigenvalue C=Mean crossings
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
Pro-poor growth
There are also many measures of whether income growth is pro-poor orpro-rich one drops out naturally of the Nichols (2008 2010) framework thecorrelation of mean income ui and the individual-specific growth rate ri Estimated in Nichols and Rehm (2014)
FranceSwitzerland
ItalyAustria
DenmarkGermany
Great BritainUnited States
SpainAustralia
LuxembourgCanadaSwedenFinland
BelgiumGreeceEstonia
KoreaPortugal
IrelandIcelandNorway
-4 -2 0 2 4 6 8 1Correlation of Mean (Longitudinal) Incomes of Individuals and Growth Rates of Individual Incomes
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Other measuresGE(2) measureGE(2) estimatesComparing across measuresFurther reading
See also
The book by Corak (2006) contains work by many authors touchpoints for alot of the subsequent work on economic mobility across generations Onmeasuring mobility Fields and Ok (1999) and Fields (2007) have a lot of ideasabout how it should be done On measuring poverty see Jenkins (2006) andon measuring inequality start with Cowell (2011) and Jenkins (2009) Some ofmy own work on poverty income inequality and mobility is accessible at
I httpppedorg
I httpwwwurbanorgeconomyEconomic-Insecuritycfm
I httpwwwurbanorginequality
I httpwwwurbanorgeconomicmobility
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Burkhauser Richard V and John G Poupore (1997) ldquoA Cross-National ComparisonOf Permanent Inequality In The United States And Germanyrdquo The Review ofEconomics and Statistics 79(1) 10ndash17
Corak Miles (editor) (2006) Generational Income Mobility in North America andEurope Cambridge Cambridge University Press
Cowell Frank A (2011) Measuring Inequality 3rd edition London School ofEconomics Perspectives in Economic Analysis
Dardanoni Valentino (1993) ldquoOn measuring social mobilityrdquo Journal of EconomicTheory 61 372ndash394
Fields Gary S and Efe Ok (1999) ldquoThe measurement of income mobilityrdquo In JSilber (Ed) Handbook of Income Distribution Measurement Boston MA Kluwer
Fields Gary S (2007) ldquoDoes income mobility equalize longer-term incomes Newmeasures of an old conceptrdquo Retrieved 17 Oct 2008 from Cornell University ILRSchool site
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Jenkins Stephen P (2006) ldquoEstimation and interpretation of measures of inequalitypoverty and social welfare using Statardquo North American Stata Usersrsquo Group Meetingrevised 6 December 2008
Jenkins Stephen P (2009) ldquoThe measurement of economic inequalityrdquo In WeimerSalverda Brian Nolan and Smeeding Timothy M (eds) The Oxford Handbook ofEconomic Inequality Oxford University Press Oxford UK 40ndash67
Kopczuk Wojciech Emmanuel Saez and Jae Song (2010) ldquoEarnings Inequality andMobility in the United States Evidence from Social Security Data since 1937rdquo TheQuarterly Journal of Economics 125(1) 91-128
Maasoumi Esfandiar (1986) ldquoThe Measurement and Decomposition ofMulti-dimensional Inequalityrdquo Econometrica Econometric Society 54(4) 991ndash97
Maasoumi Esfandiar and Mark Trede (2001) ldquoComparing Income Mobility InGermany And The United States Using Generalized Entropy Mobility Measuresrdquo TheReview of Economics and Statistics 83(3) 551ndash559
Maasoumi Esfandiar and Sourushe Zandvakili (1986) ldquoA class of generalizedmeasures of mobility with applicationsrdquo Economics Letters 22(1) 97ndash102
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Nichols Austin (2007) ldquoCausal inference with observational datardquo The StataJournal 7(4) 507ndash541
Nichols Austin (2008) ldquoTrends in Income Inequality Volatility and Mobility RiskrdquoIRISS Working Paper 94
Nichols Austin (2010) ldquoIncome Inequality Volatility and Mobility Risk in China andthe USrdquo China Economic Review 21(S1) S3ndashS11
Nichols Austin and Melissa M Favreault (2009) ldquoA Detailed Picture ofIntergenerational Transmission of Human Capitalrdquo Washington DC Urban Institute
Nichols Austin and Philipp Rehm (2014) ldquoIncome Risk in 30 Countriesrdquo Review ofIncome and Wealth 60(S1) S98ndashS116
Austin Nichols Measuring mobility
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-
Transition matricesMatrix-based mobility measures
Other mobility measuresReferences
Shorrocks Anthony F (1978) ldquoIncome Inequality and Income Mobilityrdquo Journal ofEconomic Theory 19 376-93
Shorrocks Anthony F (1978) ldquoThe Measurement of Mobilityrdquo Econometrica 461013-24
Shorrocks Anthony F (1980) ldquoThe Class of Additively Decomposable InequalityMeasuresrdquo Econometrica 48(3) 613-625
Shorrocks Anthony F (1982) ldquoInequality decomposition by factor componentsrdquoEconometrica 50 193-211
Shorrocks Anthony F (1984) ldquoInequality decomposition by population subgroupsrdquoEconometrica 52 1369-85
Austin Nichols Measuring mobility
- Transition matrices
-
- Transition matrices
- Alternatives
- Quantile transition matrices
- Markov matrices
- Estimation
-
- Matrix-based mobility measures
-
- Transition matrices
- Mobility measures
-
- Other mobility measures
-
- Other measures
- GE(2) measure
- GE(2) estimates
- Comparing across measures
- Further reading
-
- References
-