Part 11: Heterogeneity [ 1/36]

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business

Part 11: Heterogeneity [ 2/36]

Agenda Random Parameter Models

Fixed effects Random effects

Heterogeneity in Dynamic Panels Random Coefficient Vectors-Classical vs.

Bayesian General RPM Swamy/Hsiao/Hildreth/Houck Hierarchical and “Two Step” Models ‘True’ Random Parameter Variation

Discrete – Latent Class Continuous

Classical Bayesian

Part 11: Heterogeneity [ 3/36]

A Capital Asset Pricing Model

2it 0t 1t i 2t i 3t i it

it

0t

1t Mt 0t

2t

R s

R one period percentage return

expected return on a riskless security (stochastic)

expected premium on the 'market' portfolio, R R

"nonline

3t

2it i i i

ar" risk effect

"nonbeta risk" term

Data are [R , , ,s ], generated by auxiliary regressions

Coefficients are 'random' through time.

Fama - MacBeth, "Risk, Return, and Equilibrium: Empirical

Tes

ts," Journal of Political Economy, 1974.

Part 11: Heterogeneity [ 4/36]

Heterogeneous Production Model

i,t i i i,t i i,t i,tHealth HEXP EDUC

i country, t=year

Health = health care outcome, e.g., life expectancy

HEXP = health care expenditure

EDUC = education

Parameter heterogeneity:

Discrete? Aids domin

ated vs. QOL dominated

Continuous? Cross cultural heterogeneity

World Health Organization, "The 2000 World Health Report"

Part 11: Heterogeneity [ 5/36]

Parameter Heterogeneity

it i it

it i it

i i

i i

i i

X i i

i i

y c

y

u ,

E[u | ] 0 --> Random effects

E[u | ] 0 --> Fixed effects

E E[u | ] 0.

Var[u | ] not yet defined -

it

it

Unobserved Effects Random Constants

x β

x β

X

X

X

X so far, constant.

Part 11: Heterogeneity [ 6/36]

Parameter Heterogeneity

it it

i

i i

X i i

i i

y

E[ | ] zero or nonzero - to be defined

E [E[ | ]] =

Var[ | ] to be defined, constant or variable

it i

i

Generalize to Random Parameters

x β

β β u

u X

u X 0

u X

"The Pooling Problem:" What is the consequence

of estimating under the erroneous assumption of

constant parameters. (Theil, 1960, "The Aggregation

Problem") (Maddala, 1970s-1990s, "The Pooling

Problem")

Part 11: Heterogeneity [ 7/36]

Fixed Effects (Hildreth, Houck, Hsiao, Swamy)

it it

i i

i

i

i i i

i i i i

X i i X i i

i

y , each observation

, T observations

Assume (temporarily) T > K.

E[ | ] =g( ) (conditional mean)

P[ | ] =( -E[ ]) (projection)

E [E[ | ]] = E [P[ | ]] =

Var[ |

it i

i i i

i

x β

y Xβ ε

β β u

u X X

u X X X θ

u X u X 0

u Xi] constant but nonzeroΓ

Part 11: Heterogeneity [ 8/36]

OLS and GLS Are Inconsistent

i i

i

i i i i

i i

i i i i i i

, T observations

, T observations

E[ | ] E[ | ] E[ | ]

i i i

i

i i

i

i

y Xβ ε

β β u

y Xβ Xu ε

y Xβ w

w X X u X ε X 0

Part 11: Heterogeneity [ 9/36]

Estimating the Fixed Effects Model

Ni 1

Estimator: Equation by equation OLS or (F)GLS

1 ˆEstimate ? is consistent N

1 1 1 1

2 2 2 2

N N N N

i

y X 0 ... 0 β ε

y 0 X ... 0 β ε

... ... ... ... ... ... ...

y 0 0 ... X β ε

β β for E[ ] in N.iβ

Part 11: Heterogeneity [ 10/36]

Partial Fixed Effects Model

i i

N 1 N 1i 1 i 1 i

1i

Some individual specific parameters

+ , T observations

Use OLS and Frisch-Waugh

ˆ [ ] [ ], I ( )

ˆˆ [ ] ( - )

E.g., Individual specific tim

i i i i

i i ii D i i D i D i i i

i i i i

y Dα Xβ ε

β XM X XM y M D DD D

α DD D y Xβ

it i0 i1 it

it i0 it

e trends,

y t ; Detrend individual data, then OLS

E.g., Individual specific constant terms,

y ; Individual group mean deviations, then OLS

it

it

x β

x β

Part 11: Heterogeneity [ 11/36]

Heterogeneous Dynamic Models

i,t i i i,t 1 i it i,t

ii

i

logY logY x

long run effect of interest is 1

See:

Pesaran,H.,Smith,R.,Im,K.,"Estimating Long-Run Relationships

From Dynamic Heterogeneous Panels," Journal of Econometrics,

1995.

(Repeated with further study in Matyas and Sevestre, The

Econometrics of Panel Data.

Smith, J ., notes, Applied Econometrics, Dynamic Panel Data Models,

University of Warwick.

http://www2.warwick.ac.uk/fac/soc/economics/staff/faculty/jennifersmith/panel/

Weinhold, D., "A Dynamic "Fixed Effects" Model for Heterogeneous

Panel Data," London School of Economics, 1999.

Part 11: Heterogeneity [ 12/36]

Random Effects and Random Parameters

it it

i i

i

i

i i

i i

Random Parameters Model

y , each observation

, T observations

Assume (temporarily) T > K.

E[ | ] =

Var[ | ] constant but nonzero

We differentiate the classical and

it i

i i i

i

THE

x β

y Xβ ε

β β u

u X 0

u X Γ

Bayesian interpretations

Randomness here is heterogeneity, not "uncertainty"

Bayesian approach to be considered later.

Part 11: Heterogeneity [ 13/36]

Estimating the Random Parameters Model

i i

i

i i i i

i i

i i i i i i

2 2i i i i ,i ,i

2,i

, T observations

, T observations

E[ | ] E[ | ] E[ | ]

Var[ | ] Should vary by i?

,

i i i

i

i i

i

i

y Xβ ε

β β u

y Xβ Xu ε

y Xβ w

w X X u X ε X 0

w X XΓX I <==

Objects of estimation: β, Γ

Second le ivel estimation: β

Part 11: Heterogeneity [ 14/36]

Estimating the Random Parameters Model by OLS

i i

i

i i i i

i i

N 1 N N 1 Ni 1 i i i 1 i i i 1 i i i 1 i i

N 1 N 2 N 1i 1 i i i 1 i i i i 1 i i

2i

, T observations

, T observations

[ ] [ ] [ ] [ ]

[ ]=[ ] [ ( ) ][ ]

= [

i i i

i

i i

i

y Xβ ε

β β u

y Xβ Xu ε

y Xβ w

b XX Xy β XX Xw

Var b| X XX X XΓX I X XXN 1 N 1 N N 1

1 i i i 1 i i i 1 i i i i 1 i i

N 1 N N 1i 1 i i i 1 i i i i i 1 i i

] [ ] [ ( ) ( )][ ]

the usual + the variation due to the random parameters

Robust estimator

ˆ ˆEst.Var[ ] [ ] [ ][ ]

XX XX XX Γ XX XX

b XX Xw w X XX

Part 11: Heterogeneity [ 15/36]

Estimating the Random Parameters Model by GLS

i i

i

i i i i

2i i i i i i ,i

N 1 Ni 1 i i i 1 i i

2,i

, T observations

, T observations

, Var[ | ] = =( )

ˆ [ ] [ ]

ˆFor FGLS, we need and .ˆ

i i i

i

i i

i i

-1 -1i i

y Xβ ε

β β u

y Xβ Xu ε

y Xβ w w X Ω XΓX I

β XΩ X XΩ y

Γ

Part 11: Heterogeneity [ 16/36]

Estimating the RPM

i

1

1

2 1,i

T 22 t 1 it,i

i

i

( ) , = +

= ( )

Var[ | ]= + ( )

(y ) is unbiasedˆ

T K

(but not consistent because T is fixed).

i i i i i i i i i

i i i i i

i i i i

it i

b β X X X w w Xu ε

β u X X Xε

b X Γ X X

x b

Part 11: Heterogeneity [ 17/36]

An Estimator for Γ

2 1,i

X X

2 1X ,i

2 1X ,i

Ni 1

E[ | ]

Var[ | ]= + ( )

Var[ ] Var E[ | ] E Var[ | ]

= 0+ E [ + ( ) ]

+E [ ( ) ]

1Estimate Var[ ] with (

N

i i

i i i i

i i i i i

i i

i i

i

b X β

b X Γ X X

b b X b X

Γ X X

Γ X X

b

2 1 N 2 1X ,i i 1 ,i

N N 2 1i 1 i 1 ,i

)( )

1EstimateE [ ( ) ] with ( )ˆ

N1 1ˆ= ( )( ) ( )ˆN N

i i

i i i i

i i i i

b b b b '

X X X X

Γ b b b b ' - X X

Part 11: Heterogeneity [ 18/36]

A Positive Definite Estimator for Γ

N N 2 1

i 1 i 1 ,i

i

1 1ˆ= ( )( ) - ( )ˆN N

May not be positive definite. What to do?

(1) The second term converges (in theory) to 0 in T. Drop it.

(2) Various Bayesian "shrinkage" estimators,

i i i iΓ b b b b ' X X

(3) An ML estimator

Part 11: Heterogeneity [ 19/36]

Estimating βi

Ni 1

N 2 1 1 2 1i 1 ,i ,i

2 1 1 1,i

ˆ

{ [ ( ) ]} [ ( ) ]

Best linear unbiased predictor based on GLS is

ˆ ˆ ˆ + ( - ) ( )

{ [ ( ) ] }

GLS i i,OLS

i i i i i

i i GLS i i,OLS i,OLS i GLS i,OLS

-1i i i

β Wb

W Γ X X Γ X X

β Aβ I A b b A β b

A Γ X X Γ

ˆ ˆVar[ | all data]= Var[ ]

ˆVar[ ] Var[ ] [ ( - )]

( - )Var[ ] Var[ ]

-1

i i GLS i

GLS i,OLS i ii i

ii,OLS i i,OLS

β A β A

β b W AA I A

I AW b b

Part 11: Heterogeneity [ 20/36]

Baltagi and Griffin’s Gasoline Data

World Gasoline Demand Data, 18 OECD Countries, 19 yearsVariables in the file are

COUNTRY = name of country YEAR = year, 1960-1978LGASPCAR = log of consumption per carLINCOMEP = log of per capita incomeLRPMG = log of real price of gasoline LCARPCAP = log of per capita number of cars

See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp. 117-137.  The data were downloaded from the website for Baltagi's text.

Part 11: Heterogeneity [ 21/36]

OLS and FGLS Estimates

+----------------------------------------------------+| Overall OLS results for pooled sample. || Residuals Sum of squares = 14.90436 || Standard error of e = .2099898 || Fit R-squared = .8549355 |+----------------------------------------------------++---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ Constant 2.39132562 .11693429 20.450 .0000 LINCOMEP .88996166 .03580581 24.855 .0000 LRPMG -.89179791 .03031474 -29.418 .0000 LCARPCAP -.76337275 .01860830 -41.023 .0000+------------------------------------------------+| Random Coefficients Model || Residual standard deviation = .3498 || R squared = .5976 || Chi-squared for homogeneity test = 22202.43 || Degrees of freedom = 68 || Probability value for chi-squared= .000000 |+------------------------------------------------++---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ CONSTANT 2.40548802 .55014979 4.372 .0000 LINCOMEP .39314902 .11729448 3.352 .0008 LRPMG -.24988767 .04372201 -5.715 .0000 LCARPCAP -.44820927 .05416460 -8.275 .0000

Part 11: Heterogeneity [ 22/36]

Country Specific Estimates

Part 11: Heterogeneity [ 23/36]

Estimated Γ

Part 11: Heterogeneity [ 24/36]

Two Step Estimation (Saxonhouse)

it i it

i

i

it it

i i i

A Fixed Effects Model

y

Secondary Model

Two approaches

(1) Reduced form is a linear model with time constant z

y

(2) Two step

(a) FEM at step 1

(b) a (a

it

i

it i

x β

zδ

x β zδ

i i

2 1i i

i

i

) v

1 Var[v ] ( )

T

Use weighted least squares regression of a on

i

ii D i i

i

zδ

x XM X x

z

Part 11: Heterogeneity [ 25/36]

A Hierarchical Model

it i it

i i

i

it i it

i

Fixed Effects Model

y

Secondary Model

u <========

Two approaches

(1) Reduced form is an REM with time constant z

y u

(2) Two step

(a) FEM at step 1

(b) a

it

i

it i

x β

zδ

x β zδ

i i i i i

2 2 1i i u i

i

(a ) u v

1 Var[u v ] ( )

T

i

ii D i i

zδ

x XM X x

Part 11: Heterogeneity [ 26/36]

Analysis of Fannie Mae Fannie Mae The Funding Advantage The Pass Through

Passmore, W., Sherlund, S., Burgess, G.,“The Effect of Housing Government-SponsoredEnterprises on Mortgage Rates,” 2005,Federal Reserve Board and Real Estate

Economics

Part 11: Heterogeneity [ 27/36]

Two Step Analysis of Fannie-Mae

0 1 2 3i,s,t s,t s,t s,t i,s,t s,t i,s,t

4 5s,t i,s,t s,t i,s,t s,t i,s,t i,s,t

Fannie Mae's GSE Funding Advantage and Pass Through

RM ( LTV) Small Fees

New MtgCo J

i,s, t individual,state,month

1,036,252 observations in 370 state,months.

RM mortgage

LTV= 3 dummy variables for loan to value

Small = dummy variable for small loan

Fees = dummy variable for whether fees paid up front

New = dummy varia

ble for new home

MtgCo = dummy variable for mortgage company

J = dummy variable for whether this is a JUMBO loan

THIS IS THE COEFFICIENT OF INTEREST.

Part 11: Heterogeneity [ 28/36]

Average of 370 First Step RegressionsSymbol Variable Mean S.D. Coeff S.E.

RM Rate % 7.23 0.79

J Jumbo 0.06 0.23 0.16 0.05

LTV1 75%-80% 0.36 0.48 0.04 0.04

LTV2 81%-90% 0.15 0.35 0.17 0.05

LTV3 >90% 0.22 0.41 0.15 0.04

New New Home

0.17 0.38 0.05 0.04

Small < $100,000

0.27 0.44 0.14 0.04

Fees Fees paid 0.62 0.52 0.06 0.03

MtgCo Mtg. Co. 0.67 0.47 0.12 0.05

R2 = 0.77

Part 11: Heterogeneity [ 29/36]

Second Step

s,t 0

1 s,t

2 s,t

3 s,t

4 s,t

5 s,t

6

GSE Funding Advantage - estimated separately

Risk free cost of credit

Corporate debt spreads - estimated 4 different ways

Prepayment spread

Maturity mismatch risk

A

s,t

7 s,t

8 s,t

9 s,t

10-13 s,t

14-16 s,t

ggregate Demand

Long term interest rate

Market Capacity

Time trend

4 dummy variables for CA, NJ , MD, VA

3 dummy variables for calendar quarters

Part 11: Heterogeneity [ 30/36]

Estimates of β1

Second step based on 370 observations. Corrected for

"heteroscedasticity, autocorrelation, and monthly clustering."

Four estimates based on different estimates of corporate

credit spread:

0.07 (0.11) 0

11 1

21 11 2 3 4

1 1 1 1 1 1 1 1 31 1

41 1

.31 (0.11) 017 (0.10) 0.10 (0.11)

Reconcile the 4 estimates with a minimum distance estimator

ˆ( - )

ˆ( - )ˆˆ ˆ ˆ ˆMinimize [( - ),( - ),( - ),( - )]'ˆ( - )

ˆ( - )

-1Ω

Estimated mortgage rate reduction: About 7 basis points. .07%.

Part 11: Heterogeneity [ 31/36]

A Hierarchical Linear Model

German Health DataHsat = β1 + β2AGEit + γi EDUCit + β4 MARRIEDit + εit

γi = α1 + α2FEMALEi + ui

Sample ; all$Reject ; _Groupti < 7 $Regress ; Lhs = newhsat ; Rhs = one,age,educ,married ; RPM = female ; Fcn = educ(n) ; pts = 25 ; halton ; pds = _groupti ; Parameters$Sample ; 1 – 887 $Create ; betaeduc = beta_i $Dstat ; rhs = betaeduc $Histogram ; Rhs = betaeduc $

Part 11: Heterogeneity [ 32/36]

OLS Results

OLS Starting values for random parameters model...Ordinary least squares regression ............LHS=NEWHSAT Mean = 6.69641 Standard deviation = 2.26003 Number of observs. = 6209Model size Parameters = 4 Degrees of freedom = 6205Residuals Sum of squares = 29671.89461 Standard error of e = 2.18676Fit R-squared = .06424 Adjusted R-squared = .06378Model test F[ 3, 6205] (prob) = 142.0(.0000)--------+--------------------------------------------------------- | Standard Prob. Mean NEWHSAT| Coefficient Error z z>|Z| of X--------+---------------------------------------------------------Constant| 7.02769*** .22099 31.80 .0000 AGE| -.04882*** .00307 -15.90 .0000 44.3352 MARRIED| .29664*** .07701 3.85 .0001 .84539 EDUC| .14464*** .01331 10.87 .0000 10.9409--------+---------------------------------------------------------

Part 11: Heterogeneity [ 33/36]

Maximum Simulated LikelihoodNormal exit: 27 iterations. Status=0. F= 12584.28------------------------------------------------------------------Random Coefficients LinearRg ModelDependent variable NEWHSATLog likelihood function -12583.74717Estimation based on N = 6209, K = 7Unbalanced panel has 887 individualsLINEAR regression modelSimulation based on 25 Halton draws--------+--------------------------------------------------------- | Standard Prob. Mean NEWHSAT| Coefficient Error z z>|Z| of X--------+--------------------------------------------------------- |Nonrandom parametersConstant| 7.34576*** .15415 47.65 .0000 AGE| -.05878*** .00206 -28.56 .0000 44.3352 MARRIED| .23427*** .05034 4.65 .0000 .84539 |Means for random parameters EDUC| .16580*** .00951 17.43 .0000 10.9409 |Scale parameters for dists. of random parameters EDUC| 1.86831*** .00179 1044.68 .0000 |Heterogeneity in the means of random parameterscEDU_FEM| -.03493*** .00379 -9.21 .0000 |Variance parameter given is sigmaStd.Dev.| 1.58877*** .00954 166.45 .0000--------+---------------------------------------------------------

Part 11: Heterogeneity [ 34/36]

“Individual Coefficients”--> Sample ; 1 - 887 $--> create ; betaeduc = beta_i $--> dstat ; rhs = betaeduc $Descriptive StatisticsAll results based on nonmissing observations.==============================================================================Variable Mean Std.Dev. Minimum Maximum Cases Missing==============================================================================All observations in current sample--------+---------------------------------------------------------------------BETAEDUC| .161184 .132334 -.268006 .506677 887 0

Fre

qu

en

cy

BETAEDUC

-.268 -.157 -.047 .064 .175 .285 .396 .507

Part 11: Heterogeneity [ 35/36]

A Hierarchical Linear Model

A hedonic model of house values Beron, K., Murdoch, J., Thayer, M.,

“Hierarchical Linear Models with Application to Air Pollution in the South Coast Air Basin,” American Journal of Agricultural Economics, 81, 5, 1999.

Part 11: Heterogeneity [ 36/36]

HLM

ijk

M m mijk jk ijk ijkm 1

mijk

y log of home sale price i, neighborhood j, community k.

y x (linear regression model)

x sq.ft, #baths, lot size, central heat, AC, pool, good view,

age, distance to b

m

qm

Qm q qjk j jk jkq 1

qjk

Sq s qmj j js 1

qmj

each

Random coefficients

N w

N %population poor, race mix, avg age, avg. travel to work,

FBI crime index, school avg. CA achievement test score

E v

E air qu

ality measure, visibility