finding quantile gains of movers with selection …econseminar/seminar2016/dec21.lee.pdfthe...

Finding Quantile Gains of Movers with SelectionCorrection under Heteroskedasticity and

Hetero-Correlation

Jin-Young Choi+ and Myoung-Jae Lee

University of Frankfurt+ and Korea University

November 25, 2016

Jin-Young Choi+ and Myoung-Jae Lee (University of Frankfurt+ and Korea University)Finding Quantile Gains of Movers with Selection Correction under Heteroskedasticity and Hetero-CorrelationNovember 25, 2016 1 / 23

Motivation

In sample selection models, the interest is in a potentialY , but Y isobserved only for a subpopulation with D = 1. Learning about Y for thewhole population using only data from the D = 1 subgroup is the main goal.

The usual remedy is adding a selection correction term, which is a nuisanceentity without interest on its own. But there are cases where selectioncorrection terms are of genuine interest.

E.g., the study population is the unemployed at a baseline time, and overtime, some nd a job (D = 1), and the rest dont (D = 0). The interest is inthe potential wage Y at the state 1 (employed) v. state 0 (unemployed).

Typically, those getting employed is not a random sample. The potential gainE (Y jD = 1) E (Y jD = 0) is of interest, although Y is observed onlywhen D = 1.

Another example is the gain from migration, where the study population ofinterest is the source country people, Y is the potential wage in the hostcountry which is observed only when one migrates (D = 1).


Causal Analysis?

If causal analysismeans comparing two potential responses, say Y 1

(treated) and Y 0 (untreated), then our analysis is not causal.

In the wage-work example, there is no potential wage Y 0 for not working(D = 0); D is not a treatment. Y 1 exists only for working (D = 1), and ourinterest is on E (Y 1 jD = 1) E (Y 1 jD = 0) with Y = Y 1.

In the migration example, D can be construed as a treatment. Under theassumption that everybody works, Y 1 is the potential wage in the hostcountry and Y 0 is the potential wage in the source country.

In the migration example, our interest is on E (Y 1 jD = 1) E (Y 1 jD = 0):the host country potential wage di¤erence when everybody in the sourcecountry migrated, contrary to the fact. Y 0 does not appear here.

In contrast, E (Y 1 jD = 1) E (Y 0 jD = 1) is the potential wage di¤erenceof the movers in the host and source country; this is causal. Also interestingwould be E (Y 0 jD = 1) E (Y 0 jD = 0) when everybody stayed.


Goal: Find Quantile Gains

For regressors X , let qτ(Y jX ) denote the τ quantile of Y jX , 0 < τ < 1.

Our goal is to nd the X -conditional quantile gainsof the movers:

gain relative to the stayers : qτ(Y jX ,D = 1) qτ(Y jX ,D = 0);gain relative to the population : qτ(Y jX ,D = 1) qτ(Y jX ).

The former is preferred because the movers and stayers are mixed inqτ(Y jX ) of the latter. Integrating X out gives the marginal gains. We ndquantile gains, using quantile selection correction terms.

We can learn more by looking at the quantiles, not just at the mean.Di¤erently from E (jX ), however, qτ(jX ) is not a linear functional, whichmakes attaining our goal di¢ cult.


Literature Review 1

Estimators for selection: Newey et al. (1990, AER P&P), Ahn & Powell(1993, JOE), Lewbel (2007, JOE), Chen & Zhou (2010, JOE), Escanciano etal. (2014, JOE), Escanciano & Chu (2015, ER), Lee (2017, JRSS-A), ...

These estimators are, however, not helpful for nding quantile selectioncorrection terms, because they are for removing those correction terms, orthey are mean-based/too-restrictive.

The sole exception is Buchinsky (1998, JAE) who showed how to constructsemiparametric quantile selection correction terms using a single-index modelfor the selection equation.

Buchinsky (1998) did a series approximation for the correction; the interceptin the approximation is mixed with that of the outcome regression X 0β.

To separate the two intercepts, Buchinsky (1998) invoked identication atinnity requiring a known subpopulation with P(D = 1jX ) = 1.

A survey on quantile sample selection is Arellano & Bonhomme (2017,Handbook of Quantile Regression).


Literature Review 2

The intercept and slopes of β in quantile regression change when the quantilelevel τ does, if there is an ignored error term heteroskedasticity. When it isaccounted for, however, only the intercept in β changes.

The outcome equation slopes remain the same for di¤erent τs in Buchinsky(1998), because the error term heteroskedasticity through the selectionequation single-index is accounted for.

Huber & Melly (2015) took advantage of this to devise a test forindependence between the two errors. The slopes remain also the same in ourapproach (no index restriction), as the heteroskedasticity is accounted for.

Buchinsky (1998) has no provision for hetero-correlationbetween the twoerrors, but it seems allowed through the single index. Hetero-correlation israrely considered, other than briey in Arellano & Bonhomme (ECA, 2017).

Arellano & Bonhomme allowed a exible parametric distribution of the twoerrors with a copula indexed by correlation ρ, and estimated quantileparameters βτ (heteroskedasticity allowed), not subject to selection bias.


Our Approach and Main Contribution

We propose a fully parametric MLE-based quantile selection correctionapproach, under the joint normality between the two errors.

We allow heteroskedasticity and hetero-correlation of known forms: anexponential specication for the former, and a smooth function bounded by1 for the latter. Both heterogeneity terms matter greatly in our application.

The selection correction terms are obtained by a simulation-basedclosed-formapproach no optimization beyond the MLE. Our proposal isstraightforward to implement with no tuning parameter to choose.

The most important contribution of our approach is enabling the comparisonbetween the movers and the stayers, whereas Buchinsky (1998, JAE) cancompare the movers only to the population.

Comparing the movers to the population is misleading when the populationconsists mostly of movers for a given X , because the quantile gain relative tothe population becomes automatically almost zero.


Base Selection Model

For regressors X Z , the selection model is

Di = 1[0 < Z0i α+ εi ], Y i = X

0i β+ Ui ,

Yi = Yi Di ; (Zi ,Yi ), i = 1, ...,N, observed. (2.1)

Assume εU

N(0,Ω)q Z , Ω =

1 σεu

σεu σ2u

=

1 ρσu

ρσu σ2u

,

σu SD(U), σεu COV (ε,U) and ρ COR(ε,U). (2.2)

(2.2) implies, for some ν N(0, 1)q ε,

U = σεu ε+ σν = σu (ρε+q1 ρ2ν), σ σu

q1 ρ2; (2.3)

U/σu = (ρ2)1/2ε+ (1 ρ2)1/2ν is easier to remember.


MLE and Quantile Correction Term

With λ (Z 0α) φ(Z 0α)/Φ(Z 0α) = E (εjZ ,D = 1),

E (U jZ ,D = 1) = ρσuλ(Z 0α) =) E (UσujZ ,D = 1) = ρλ(Z 0α).

Because qτ(U jZ ,D = 1) = σuqτ(U/σu jZ ,D = 1) and U/σu N(0, 1),qτ(Y jZ ,D = 1) = X 0β+ σuqτfN(0, 1)jZ ,D = 1g.

First, obtain the full MLE (α, β, ρ, σu); second, nd qτfN(0, 1)jZ ,D = 1)with a simple simulation approach explained shortly; third, obtain

qτ(U jZ ,D = 1) = σu qτfN(0, 1)jZ ,D = 1g.MLE maximizes

ln L(α, β, ρ, σu) N

∑i=1[ (1Di ) lnΦ(Z 0i α)

+Di [lnΦf(Z 0i α+ ρYi X 0i β

σu)(1 ρ2)1/2g ln σu + ln φ(

Yi X 0i βσu

)] ].


Simulation Estimation of Conditional Quantiles for N(0,1)

Dene

qZ1τ qτ(UσujZ ,D = 1) and qZ0τ qτ(

UσujZ ,D = 0)

and observe qτ(U jZi , ε > Z 0i α) = σu qZi1τ.

With (α, β, ρ, σu) in hand, generate for each i

εj ,Vj iid from N(0, 1) and Uj ρεj +

q1 ρ2V j , j = 1, ..., ni .

Choose Uj s with εj > Z 0i α, and obtain the τ quantile qZi1τ among thechosen Uj s. Compare this to E (U jZi , ε > Z 0i α) estimated by the mean ofthe Ujs with εj > Z 0i α, which is a simulation estimation of λ(Z 0i α).

We can estimate qZi0τ analogously by choosing Uj s with εj Z 0i α and then

getting the τ quantile.


Marginal Quantile Gains

Dene marginal quantile gains of movers(G1τ is preferred to G0τ):

G1τ Efqτ(U jZ ,D = 1) qτ(U jZ ,D = 0)jD = 1g,G0τ Efqτ(U jZ ,D = 1) qτ(U jZ )jD = 1g.

G1τ and G0τ can be estimated by (N1 is the number of movers)

G1τ 1N1

∑i2fD=1g

σu(qZi1τ q

Zi0τ) & G0τ

1N1

∑i2fD=1g

σufqZi1τ qτ(N(0, 1))g

To see G1τ v. G0τ better, consider the mean version of G0τ ignoring Z :

E (U jD = 1) E (U)= E (U jD = 1) E (U jD = 1)P(D = 1) E (U jD = 0)P(D = 0)= fE (U jD = 1) E (U jD = 0)g P(D = 0).

E (U jD = 1) E (U) dilutes E (U jD = 1) E (U jD = 0) to the extentP(D = 0) < 1. The same would be true of G0τ v. G1τ.


Heteroskedasticity of Known Form

To allow heteroskedasticity, replace (2.3) with

U = σu(X ) (ρε+q1 ρ2ν) and σu(X ) = exp(X 0µ);

no problem in using σu(Z ) instead of σu(X ).

Since U/σu(X ) N(0, 1) in

qτ(Y jZ ,D = 1) = X 0β+ σu(X ) qτfU

σu(X )jZ ,D = 1g,

use the same qZi1τ as before.

Thenqτ(U jZi ,D = 1) = exp(X 0i µ) q

Zi1τ.

The marginal gains can be estimated with

1N1

∑i2fD=1g

exp(X 0i µ)(qZi1τ q

Zi0τ),

1N1

∑i2fD=1g

exp(X 0i µ)fqZi1τ qτ(N(0, 1))g.


Hetero-Correlation of Known Form

To allow ρ to depend on Z , replace (2.3) with

U = σu fρ(Z )ε+q1 ρ2(Z )νg and ρ(Z ) = ssn(Z 0η) 2Φ(Z 0η) 1.

2Φ(Z 0η) 1 is a smoothed version of the sign function 2 1[Z 0η > 0] 1;ssnstands for smoothed sign. Certainly, ρ(Z ) can be modeled di¤erently.

For simulation-based estimation after MLE, generate, for each Z 0i α and Z0i η,

εj ,Vj iid from N(0, 1), Uj ssn(Z 0i η)εj +

q1 ssn2(Z 0i η)V

j , j = 1, ..., ni .

The marginal gains can be estimated by doing as done for the base model.


Heteroskedasticity and Hetero-Correlation

To allow both σu(X ) and ρ(Z ), replace (2.3) with

U = σu(X )fρ(Z )ε+q1 ρ2(Z )νg, σu(X ) = exp(X 0µ), ρ(Z ) = ssn(Z 0η).

The log-likelihood function with parameters (α, β, η, µ) is

N

∑i=1[ (1Di ) lnΦ(Z 0i α) + Di [lnΦf(Z 0i α+ ssn(Z 0i η)

Yi X 0i βexp(X 0i µ)

)

(1 ssn2(Z 0i η))1/2g X 0i µ+ ln φ(Yi X 0i βexp(X 0i µ)

)] ].

The second component is lnfP(D = 1jU,Z ) fU jZ (u)g.P(D = 1jU,Z ) = P(ε > Z 0αjU,Z ); the mean and SD of εjU appearinside Φ() of the second component.

The simulation estimation can be done as in the hetero-correlation model,and the marginal gains can be estimated as in the heteroskedasticity model.


Empirical Analysis

Our data set is a 20% Sample of Integrated Labour Market Biographies ofGermany in 2010 (N = 112, 310). It contains daily information on individualemployment history, along with some demographics.

Y is daily wage in Euros, and D = 1 if employed. X includes job experiencesin months (Expi), cum. unemployment duration in months (Dur0),university degree (Univ), and vocational training certicate (Voc).

Also, 120 types of occupation are in X , along with seven types of job-status(part-time less than half, part-time more than half, full time, unskilled,skilled, home worker, and master (craftsman or manager)).

Although exclusion restriction is not required, the MLE does not convergewell without one. We use the ratio R-Dur0Dur0/Expi and its squaredterm as the excluded variables for the following reason.

Relative unemployment durationR-Dur0 a¤ects D, but not wage Y withDur0 and Expi already controlled for Y . This may be false, but unlikely tocause heavy heteroskedasticity and hetero-correlation in our analysis.


Descriptive Statistics

Table 1: Mean (SD) VariablesMales (N = 61053) Female (N = 51257)

Total D = 1 D = 0 Total D = 1 D = 0Y 89.0 (53) 101 (45) 62.5 (42) 70.1 (39)D 0.881 0.892Age 42.0 (11) 42.0 (11) 42.2 (12) 42.1 (11) 42.0 (11) 43.0 (11)Expi 186 (121) 196 (120) 114 (107) 164 (110) 173 (109) 97.2 (95)Dur0 17.6 (30) 11.6 (21) 62.1 (52) 17.6 (30) 12.5 (22) 58.8 (49)Univ 0.133 0.143 0.064 0.106 0.110 0.068Voc 0.702 0.710 0.637 0.741 0.751 0.664


Estimates under Hetero- skedasticity & correlation

Table 2. Regression Parameters for D and YMales (N = 61053) Females (N = 51257)α (tv) β (tv) α (tv) β (tv)

Age 0.038 (5.84) 1.27 (10.6) 0.016 (2.40) -0.213 (-1.95)

Age2/100 -0.069 (-9.40) -2.23 (-15.9) -0.038 (-4.99) -0.134 (-1.08)

Expi 0.009 (29.2) 0.246 (44.3) 0.009 (38.1) 0.169 (30.6)

Expi2/100 -0.002 (-20.8) -0.023 (-20.2) -0.002 (-19.7) -0.017 (-15.2)

Dur0 -0.031 (-66.6) -0.677 (-58.5) -0.030 (-58.9) -0.417 (-37.4)

Dur20/100 0.010 (42.1) 0.251 (26.2) 0.011 (43.0) 0.144 (17.1)

Univ 0.188 (5.34) 28.9 (46.4) 0.350 (9.62) 27.1 (38.9)

Voc 0.098 (4.59) 3.92 (9.75) 0.158 (6.87) 2.49 (6.44)

R-Dur0 -0.011 (-4.25) -0.007 (-10.4)

R-Dur20/100 0.001 (0.81) 0.0004 (2.72)


Hetero- Skedasticity & Correlation Function Estimates

Table 2. Heteroskedasticity and Hetero-CorrelationMales (N = 61053) Females (N = 51257)µ (tv) η (tv) µ (tv) η (tv)

Age 0.008 (3.76) 0.051 (3.92) 0.046 (23.7) 0.181 (11.2)

Age2/100 -0.003 (-1.32) -0.061 (-4.03) -0.046 (-20.1) -0.195 (-10.4)

Univ -0.043 (-3.73) -1.00 (-10.9) 0.314 (24.6) -1.417 (-14.2)

Voc 0.023 (2.66) 0.019 (0.39) 0.022 (2.85) -0.168 (-2.97)

Skilled -0.011 (-1.29) -0.004 (-0.07) 0.043 (2.91) -0.006 (-0.06)

Master 0.163 (6.23) -2.17 (-7.22) 0.391 (5.18) -2.82 (-3.34)

Full time 0.223 (25.6) -1.38 (-17.3) 0.329 (39.0) -1.78 (-21.3)

More than Half 0.167 (11.7) 0.284 (3.08) -0.019 (-2.31) 0.0001 (0.002)

E.g., the slope of Univ is 0.314 for females, which means100fexp(0.314) 1g% = 37% increase in SD for women due to Univ; η isharder to interpret, because ∂f2Φ(Z 0η) 1g/∂Z = 2φ(Z 0η)η.


Group Hetero- Skedasticity & Correlations

Table 3. Variance and Correlation by GroupMales Females

σu(W ) ρ(W ) σu(W ) ρ(W )Total 28.94 0.322 28.35 0.287

Univ 30.68 -0.633 38.53 -0.467

Voc 29.01 0.446 27.59 0.328

Skilled 26.02 0.781 23.80 0.794

Master 31.77 -0.648 35.07 -0.825

Full time 32.35 -0.306 33.46 -0.327

More than half 31.04 0.777 24.03 0.809

Age<30 26.00 0.401 24.50 -0.186

Age>50 30.94 0.307 28.95 0.442

Although σu(W ) does not vary much across groups, ρ(W ) does: ignoringheteroskedasticity may be innocuous, but ignoring hetero-correlation is not.


Quantile Gain Can Change as the Quantile Level Changes


Quantile Gain Relative to Population

Figure: Quantile Gain G0τ Relative to the Population

The gures reveal qτ(U jZ ,D = 1) ' qτ(U jZ ) for high τ (i.e., the right tailof U jZ distribution is lled by those with D = 1) so that

G0τ = qτ(U jZ ,D = 1) qτ(U jZ ) ' qτ(U jZ ) qτ(U jZ ) = 0.


Quantile Gain Relative to Stayers

Figure: Quantile Gain G1τ Relative to the Stayers

For low τ, qτ(U jZ ,D = 0) = qτ(U jZ ) is likely (i.e., the left tail if the U jZdistribution is lled by those with D = 0) to render G0τ ' G1τ: as τ #,G1τ G0τ drops from 22.5 to 6.8 for males, and from 18.8 to 1.3 for females.


Conclusions

We nd quantile gains in moving from one state to another, using quantileselection correction terms. Our approach is fully parametric thedisadvantage as quantiles are far more di¢ cult to deal with than the mean.

Our approach has advantages (v. the semiparametric approach of Buchinsky1998): we allow heteroskedasticity & hetero-correlation, and we can nd themover gain relative to the stayers G1τ (as well as to the population G0τ).

Using German unemployment data, we found that G1τ steadily increases asthe quantile level τ goes up. For males, G1τ = 10.5 Euros at τ = 0.1, and itincreases to 23 at τ = 0.9; for females, G1τ changes from 4.5 to 18.6.

G1τ is greater for males, and the gender gap is around 4.4 6 Euros acrossquantiles. G0τ decreases as τ goes up (while G1τ goes up), as the right tailof the U jZ dist. is lled by those with D = 1; G0τ is misleading this way.

Heteroskedasticity and hetero-correlation matter greatly; the correlationvaries widely (0.8 to 0.8) across subpopulations the most surprisingnding in our empirical analysis.


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