1/68: topic 1.3 – linear panel data regression microeconometric modeling william greene stern...

1/68: Topic 1.3 – Linear Panel Data Regression

Microeconometric Modeling

William GreeneStern School of BusinessNew York UniversityNew York NY USA

1.3 Linear Panel Data Regression Models


Concepts

• Unbalanced Panel• Cluster Estimator• Block Bootstrap• Difference in Differences• Incidental Parameters

Problem• Endogeneity• Instrumental Variable• Control Function Estimator• Mundlak Form• Correlated Random Effects• Hausman Test• Lagrange Multiplier (LM) Test• Variable Addition (Wu) Test

Models

• Linear Regression• Fixed Effects LR Model• Random Effects LR Model


BHPS Has Evolved


German Socioeconomic Panel


Balanced and Unbalanced Panels

Distinction: Balanced vs. Unbalanced Panels

A notation to help with mechanics zi,t, i = 1,…,N; t = 1,…,Ti

The role of the assumption Mathematical and notational convenience:

Balanced, n=NT Unbalanced:

The fixed Ti assumption almost never necessary.

If unbalancedness is due to nonrandom attrition from an otherwise balanced panel, then this will require special considerations.

N

ii=1n T


An Unbalanced Panel: RWM’s GSOEP Data on Health Care

N = 7,293 Households

Some households exited then returned


Cornwell and Rupert DataCornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years(Extracted from NLSY.) Variables in the file are

EXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationLWAGE = log of wage = dependent variable in regressions

These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagi's text.


Common Effects Models Unobserved individual effects in regression: E[yit | xit, ci]

Notation:

Linear specification: Fixed Effects: E[ci | Xi ] = g(Xi). Cov[xit,ci] ≠0 effects are correlated with included variables.

Random Effects: E[ci | Xi ] = μ; effects are uncorrelated with included variables. If Xi contains a constant term, μ=0 WLOG. Common: Cov[xit,ci] =0, but E[ci | Xi ] = μ is needed for the full model

it it i ity = + c + x

i

i1

i2i i

iT

T rows, K columns

x

xX

x


Convenient Notation

Fixed Effects – the ‘dummy variable model’

Random Effects – the ‘error components model’

it i it ity = + + x

it it it iy = + + u x

Individual specific constant terms.

Compound (“composed”) disturbance


Estimating β

β is the partial effect of interest Can it be estimated (consistently) in

the presence of (unmeasured) ci? Does pooled least squares “work?” Strategies for “controlling for ci” using

the sample data.


1. The Pooled Regression

Presence of omitted effects

Potential bias/inconsistency of OLS – Depends on ‘fixed’ or ‘random’ If FE, X is endogenous: Omitted Variables Bias If RE, OLS is OK but standard errors are

incorrect.

it it i it

i i i i i

i i i i i i i

Ni=1 i

y = +c+ε , observation for person i at time t

= +c + , T observations in group i

= + + , note (c ,c ,...,c )

= + + , T observations in the sample

x β

y Xβ i ε

Xβ c ε c

y Xβ c ε


OLS with Individual EffectsThe omitted variable(s) are the group means

-1 -1

-1N Ni=1 i i i=1 i i

-1N Ni=1 i i i=1 i i

=( ) ( ) ( + + )

= + (1/N)Σ (1/N)Σ (part due to the omitted c )

+ (1/N)Σ (1/N)Σ (covariance of and will = 0)

The third term

b XX X'y= XX X' Xβ c ε

β XX Xc

XX Xε X ε

i

-1N N ii=1 i i i=1 i i

Ni=1 i i i

i i

vanishes asymptotically by assumption

T1plim = + plim Σ Σ c (left out variable formula)

N N

So, what becomes of Σ w c ?

plim = if the covariance of and c

b β XX x

x

b β x converges to zero.


Ordinary Least Squares

Standard results for OLS in a generalized regression model Consistent if RE, inconsistent if FE. Unbiased for something in either case. Inefficient in all cases.

True Variance

1 12

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

Var[ | ]T T T T

XX XΩX XXb X


Estimating the Sampling Variance of b

b may or may not be consistent for . We estimate its variance regardless

s2(X ́X)-1 is not the correct matrix Correlation across observations: Yes Heteroscedasticity: Maybe

Is there a “robust” covariance matrix? Robust estimation (in general) The White estimator for

heteroscedasticity A Robust estimator for OLS.


A Cluster Estimator

i i

i

ˆ ˆ

ˆ ˆ

ˆ

i

T TNi=1 t=1 it it t=1 it it

T TNi=1 t=1 s=1 it is it is

it

Robust variance estimator for Var[ | ]

Est.Var[ | ]

N = Σ Σ w Σ w

N-1

N = Σ Σ Σ w w

N-1

w = a least square

-1 -1

-1 -1

b X

b X

XX x x XX

XX x x XX

i

Ni=1 i

Ni=1 i

s residual.

(If T = 1, this is the White estimator.)

(The finite population correction [N/ (N-1)] is ad hoc.)

Σ T(The further finite population correction, is also ad hoc.)

Σ T N


Alternative OLS Variance EstimatorsCluster correction increases SEs

+---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ Constant 5.40159723 .04838934 111.628 .0000 EXP .04084968 .00218534 18.693 .0000 EXPSQ -.00068788 .480428D-04 -14.318 .0000 OCC -.13830480 .01480107 -9.344 .0000 SMSA .14856267 .01206772 12.311 .0000 MS .06798358 .02074599 3.277 .0010 FEM -.40020215 .02526118 -15.843 .0000 UNION .09409925 .01253203 7.509 .0000 ED .05812166 .00260039 22.351 .0000Robust Constant 5.40159723 .10156038 53.186 .0000 EXP .04084968 .00432272 9.450 .0000 EXPSQ -.00068788 .983981D-04 -6.991 .0000 OCC -.13830480 .02772631 -4.988 .0000 SMSA .14856267 .02423668 6.130 .0000 MS .06798358 .04382220 1.551 .1208 FEM -.40020215 .04961926 -8.065 .0000 UNION .09409925 .02422669 3.884 .0001 ED .05812166 .00555697 10.459 .0000


Results of Bootstrap Estimation


Bootstrap variance for a panel data estimator Panel Bootstrap =

Block Bootstrap Data set is N groups of

size Ti

Bootstrap sample is N groups of size Ti drawn with replacement.

The bootstrap replication must account for panel data nature of the data set.


Difference-in-Differences ModelWith two periods and strict exogeneity of D and T,

This is a linear regression model. If there are no regressors,

it 0 1 it 2 t 3 t it it

it

t

y = D T TD

D = dummy variable for a treatment that takes place

between time 1 and time 2 for some of the individuals,

T = a time period dummy variable, 0 in period 1,

1 in period 2.

3 t 2 t 1 D 1 t 2 t 1 D 0

Using least squares,

b (y y ) (y y )


Difference in Differences

it 0 1 it 2 t 3 it t it it

it 2 3 i2 it it

2 3 i2 it i

it it

3 it it

y = D T D T ,t 1,2

y = D ( )

= D ( ) w

y | D 1 y | D 0

( | D 1) ( | D 0)

If the effect is estimated by averaging individua

βx

βx

β x

β x x

ls with D = 1

and different individuals with D=0, then part of the 'effect'

is explained by change in the covariates, not the treatment.


UK Office of Fair Trading, May 2012; Stephen Davies

http://dera.ioe.ac.uk/14610/1/oft1416.pdf


Outcome is the fees charged.

Activity is collusion on fees.


Treatment Schools: Treatment is an intervention by the Office of Fair Trading

Control Schools were not involved in the conspiracy

Treatment is not voluntary


Apparent Impact of the Intervention


Treatment (Intervention) Effect = 1 + 2 if SS school


In order to test robustness two versions of the fixed effects model were run. The first is Ordinary Least Squares, and the second is heteroscedasticity and auto-correlation robust (HAC) standard errors in order to check for heteroscedasticity and autocorrelation.


The cumulative impact of the intervention is the area between the two paths from intervention to time T.


2. Estimation with Fixed Effects

The fixed effects model

ci is arbitrarily correlated with xit but E[εit|Xi,ci]=0

Dummy variable representation

it it i it

i i i i i

i i i i i i i

Ni=1 i

1 2 N



= + + , note (c ,c ,...,c )


c=( , ,... ) ,

x β

y Xβ i ε

Xβ c ε c

y Xβ c ε

c c c Ni=1 iT by 1 vector

Nit it j=1 j ijt it ijty = + d +ε , d = (i=j) x β 1


The Fixed Effects Model

yi = Xi + dii + εi, for each individual

1 1

2 2

N

=

=

N

1

2

N

y X d 0 0 0

y X 0 d 0 0 βε

α

y X 0 0 0 d

β [X,D] ε

α

Zδ ε

E[ci | Xi ] = g(Xi); Effects are correlated with included variables. Cov[xit,ci] ≠0


Estimating the Fixed Effects Model

The FEM is a plain vanilla regression model but with many independent variables

Least squares is unbiased, consistent, efficient, but inconvenient if N is large.

1

1

Using the Frisch-Waugh theorem

=[ ]

D D

b XX XD Xy

a DX DD Dy

b XM X XM y


The Within Transformation Removes the Effects

it it i it

i i i i

it i it i it i

it it it

y c+ε

y c+ε

y y ( ) (ε ε)

y ε

x β

xβ

x - x β

x β

Wooldridge notation for data in deviations from group means


Least Squares Dummy Variable Estimator

b is obtained by ‘within’ groups least squares (group mean deviations)

Normal equations for a are D’Xb+D’Da=D’y

a = (D’D)-1D’(y – Xb)

iTi i t=1 it it ia=(1/T)Σ (y - )=ex b

Notes: This is simple algebra – the estimator is just OLS

Least squares is an estimator, not a model. (Repeat twice.)

Note what ai is when Ti = 1. Follow this with yit-ai-xit’b=0 if Ti=1.


Inference About OLS Assume strict exogeneity: Cov[εit,(xjs,cj)]=0. Every

disturbance in every period for each person is uncorrelated with variables and effects for every person and across periods.

Now, it’s just least squares in a classical linear regression model.

Asy.Var[b] = 2 N 2 N N 1i=1 i i=1 i i=1( / T)plim[( / T) ]

which is the usual estimator for OLS

i

i D iXM X

iTN 2

2 i=1 t=1 it i itNi=1 i

(y -a-x )ˆ

T - N - K

(Note the degrees of freedom correction)

b


Application Cornwell and Rupert


LSDV Results

Pooled OLS

Note huge changes in the coefficients. SMSA and MS change signs. Significance changes completely.


Estimated Fixed Effects


The Effect of the EffectsR2 rises from .26510 to .90542


Robust Covariance Matrix for LSDVCluster Estimator for Within Estimator

Effect is less pronounced than for OLS


Endogeneity in the FEM

yi = Xi + diαi + εi for each individual

1 1

2 2

N

=

=

1

2

N

y X d 0 0 0

y X 0 d 0 0 βε

α

y X 0 0 0 d

Xβ + Dα ε

Xβ + w

N

E[wi | Xi ] = g(Xi); Effects are correlated with included variables. Cov[xit,wi] ≠0X is endogenous because of the correlation between xit and wi


i

1

LSDV

1

LSDV

i i

Tii D i t=1 it,k i.,k it,l i.,lk,l

Define = .

(Looks like an IV estimator.)

(1) Plim ?Σ T

Plim (x -x )(x -x )

Nonsingular PD matrix if ther

D D

D

b XM X XM y

Z M X

b ZX Zy

ZX0

XM X

ii i D i i

i i i i i i

ii D

e is no multicollinearity and if

every column of has within group variation.

Σ (c )(2) Plim ? Plim Plim

Σ T Σ T Σ T

c because has no within group varia

X

XM iZw Zw= 0

M i=0 i

ii i D i

i i

tion

1Plim Σ by the assumption of the model.

Σ TXM 0

The within (LSDV) estimator is an instrumental variable (IV) estimator


LSDV is a Control Function Estimator

1

We seek a control function (.) such that | (.) is uncorrelated

with . (In the presence of (.), is not correlated with .)

Using the Frisch-Waugh theorem

=[ ]

Consider regressi

D D

h X h

w h X w

b XM X XM y

on of y on [ , ]. I.e., add group means to

the regression.

X X

50/68: Topic 1.3 – Linear Panel Data RegressionLSDV is a Control Function Estimator

11 12 1K 11 1 12 1 11 1

21 22 2K 21 2 22 2 11 2

N1 N2 NK N1 N N2 N NK N

Consider regression of y on [ , ]. I.e., add group

means to the regression.

x x x

x x x[ , ]

x x x

= [

X X

x x x .i .i i

x x x .i .i iX X

x x x .i .i .i

X, ( - ) ]

= [ , ]

= [ , ]

Regression of y on [ , ] produces the LSDV estimator.

(See Appendix for a proof.)

D

D

I M X

X P X

X F

X X


The problem here is the estimator of the disturbance variance. The matrix is OK.Note, for example, .01374007/.01950085 (top panel) = .16510 /.23432 (bottom panel).


3. The Random Effects Model

The random effects model

ci is uncorrelated with xit for all t; E[ci |Xi] = 0

E[εit|Xi,ci]=0

it it i it

i i i i i

i i i i i i i

Ni=1 i

1 2 N



= + + , note (c ,c ,...,c )


c=( , ,... ) ,

x β

y Xβ i ε

Xβ c ε c

y Xβ c ε

c c c Ni=1 iT by 1 vector


Random vs. Fixed Effects

Random Effects Small number of parameters Efficient estimation Objectionable orthogonality assumption (ci Xi)

Fixed Effects Robust – generally consistent Large number of parameters More reasonable assumption Precludes time invariant regressors

Which is the more reasonable model?


Mundlak’s Estimator

ii i i i i1 i1 iT i

i

i i i i i

i i i i

i i

Write c = u , E[c | , ,... ]

Assume c contains all time invariant information


= + + + u

Looks like random effects.

Var[ + u ]=

xδ x x x = xδ

y Xβ i ε

Xβ ixδ ε i

ε i Ω +

This is the model we used for the Wu test.

2i uσ ii

Mundlak, Y., “On the Pooling of Time Series and Cross Section Data, Econometrica, 46, 1978, pp. 69-85.


A “Hierarchical” Model

it it i it

i i i

it it i it

Lower level structural model

y c+ε

Upper level model for effects

c z + w

How does this affect the fixed effects model?

y α+ε

No change in the model, but it invites a second step

x β

δ

x β

.


Mundlak’s Approach for an FE Model

with Time Invariant Variables

it it i it

2it i i it i i

i i

2i i i i w

it it i it

y + c+ε , ( does not contain a constant)

E[ε | ,c ] 0,Var[ε | ,c ]=

c + + w ,

E[w| , ] 0, Var[w| , ]

y w ε

random effe

i

i

i i

i i

x β zδ x

X X

x θ

X z X z

x β zδ x θ

cts model including group means of

time varying variables.


Correlated Random Effects

ii i i i i1 i1 iT i

i

i i i i i

i i i i

i i1 1 i2 2

c = u , E[c | , ,... ]

Assume c contains all time invariant information


= + + + u

c =

xδ x x x = xδ

y X

Mundlak

Chamberlain/ Wooldri

β i ε

Xβ ixδ ε i

x δ x

dg

δ

e

iT T i

i i i1 1 i1 2 iT T i i

... u

= ... u+

TxK TxK TxK TxK etc.

Problems: Requires balanced panels

Modern panels have large T; models have large K

x δ

y Xβ ix δ ix δ ix δ i ε


LM Tests

+--------------------------------------------------+| Random Effects Model: v(i,t) = e(i,t) + u(i) | Unbalanced Panel| Estimates: Var[e] = .216794D+02 | #(T=1) = 1525| Var[u] = .958560D+01 | #(T=2) = 1079| Corr[v(i,t),v(i,s)] = .306592 | #(T=3) = 825| Lagrange Multiplier Test vs. Model (3) = 4419.33 | #(T=4) = 926| ( 1 df, prob value = .000000) | #(T=5) = 1051| (High values of LM favor FEM/REM over CR model.) | #(T=6) = 1200| Baltagi-Li form of LM Statistic = 1618.75 | #(T=7) = 887+--------------------------------------------------++--------------------------------------------------+| Random Effects Model: v(i,t) = e(i,t) + u(i) || Estimates: Var[e] = .210257D+02 | Balanced Panel| Var[u] = .860646D+01 | T = 7| Corr[v(i,t),v(i,s)] = .290444 || Lagrange Multiplier Test vs. Model (3) = 1561.57 || ( 1 df, prob value = .000000) || (High values of LM favor FEM/REM over CR model.) || Baltagi-Li form of LM Statistic = 1561.57 |+--------------------------------------------------+

REGRESS ; Lhs=docvis ; Rhs=one,hhninc,age,female,educ ; panel $


A One Way REM


A Hausman Test for FE vs. RE

Estimator Random EffectsE[ci|Xi] = 0

Fixed EffectsE[ci|Xi] ≠ 0

FGLS (Random Effects)

Consistent and Efficient

Inconsistent

LSDV(Fixed Effects)

ConsistentInefficient

ConsistentPossibly Efficient


Hausman Test for Effects

-1

d

ˆ ˆBasis for the test,

ˆ ˆˆ ˆ ˆ ˆWald Criterion: = ; W = [Var( )]

A lemma (Hausman (1978)): Under the null hypothesis (RE)

ˆ nT[ ] N[ , ] (efficient)

ˆ nT[

FE RE

FE RE

RE RE

FE

β - β

q β - β q q q

β - β 0 V

β d] N[ , ] (inefficient)

ˆ ˆˆNote: = ( )-( ). The lemma states that in the

ˆ ˆjoint limiting distribution of nT[ ] and nT , the

limiting covariance, is . But, =

FE

FE RE

RE

Q,RE Q,RE FE,R

- β 0 V

q β - β β β

β - β q

C 0 C C - . Then,

Var[ ] = + - - . Using the lemma, = .

It follows that Var[ ]= - . Based on the preceding

ˆ ˆ ˆ ˆ ˆ ˆH=( ) [Est.Var( ) - Est.Var( )] (

E RE

FE RE FE,RE FE,RE FE,RE RE

FE RE

-1FE RE FE RE FE RE

V

q V V C C C V

q V V

β - β β β β - β )


HausmanThere is a built in procedure for this.

It is not always appropriate to compare estimators this way.


A Variable Addition Test

Asymptotically equivalent to Hausman Also equivalent to Mundlak formulation In the random effects model, using FGLS

Only applies to time varying variables Add expanded group means to the regression

(i.e., observation i,t gets same group means for all t.

Use standard F or Wald test to test for coefficients on means equal to 0. Large F or chi-squared weighs against random effects specification.


Variable Addition

it i it it

it i it i

it it i

A Fixed Effects Model

y

LSDV estimator - Deviations from group means:

To estimate , regress (y y ) on ( )

Algebraic equivalent: OLS regress y on ( )

Mundlak interpretation:

x

x x

x , x

i i i

it i i it it

i it it i

u

Model becomes y u

= u

a random effects model with the group means.

Estimate by FGLS.

x

x x

x x


Application: Wu Test

NAMELIST ; XV = exp,expsq,wks,occ,ind,south,smsa,ms,union$NAMELIST ; (new) xmeans = expb,expsqb,wksb,occb,indb,southb,smsab,msb,unionb $CREATE ; xmeans = Group Mean(xv,pds=ti) $REGRESS ; Lhs = lwage ; Rhs = xmeans,Xv,ed,fem,one ; panel ; random ; Test: xmeans $


Means Added

1/68: topic 1.3 – linear panel data regression microeconometric modeling william greene stern...

Documents

linear panel data regression2

linear panel data regression10

schooling data

regressionsthese data

sample data

linear specification

rwms gsoep data

ci notation