Econometric Analysis of Panel Data

1
Econometric Analysis of Panel Data

• William Greene
• Department of Economics
• Stern School of Business

2
The Random Effects Model

• The random effects model
• ci is uncorrelated with xit for all t
• Eci Xi 0
• EeitXi,ci0

3
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?

4
Error Components Model

• Generalized Regression Model

5
Notation
6
Notation
7
Regression Model-Orthogonality
8
Convergence of Moments
9
Ordinary Least Squares

• Standard results for OLS in a GR model
• Consistent
• Unbiased
• Inefficient
• True Variance

10
Estimating the Variance for OLS
11
Mechanics
12
Cornwell and Rupert Data
Cornwell and Rupert Returns to Schooling Data,
595 Individuals, 7 YearsVariables in the file
are EXP work experience, EXPSQ EXP2WKS
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 unioin
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.
13
Alternative OLS Variance Estimators
----------------------------------------------
---------- Variable Coefficient Standard
Error b/St.Er.PZgtz ---------------------
----------------------------------- 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
.0000 Robust 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
14
Generalized Least Squares
15
Panel Data Algebra (1)
16
Panel Data Algebra (2)
17
Panel Data Algebra (3)
18
GLS (cont.)
19
Estimators for the Variances
20
Feasible GLS
x does not contain a constant term in the
preceding.
21
Practical Problems with FGLS
22
Stata Variance Estimators
23
Computing Variance Estimators
24
Application
-------------------------------------------------
- Random Effects Model v(i,t) e(i,t) u(i)
Estimates Vare
.231188D-01 Varu
.102531D00
Corrv(i,t),v(i,s) .816006 (High
(low) values of H favor FEM (REM).)
Sum of Squares .141124D04
R-squared -.591198D00
-----------------------------------------------
--- -----------------------------------------
------------------------- Variable
Coefficient Standard Error b/St.Er.PZgtz
Mean of X ----------------------------------
-------------------------------- EXP
.08819204 .00224823 39.227 .0000
19.8537815 EXPSQ -.00076604
.496074D-04 -15.442 .0000 514.405042 OCC
-.04243576 .01298466 -3.268
.0011 .51116447 SMSA -.03404260
.01620508 -2.101 .0357 .65378151 MS
-.06708159 .01794516 -3.738
.0002 .81440576 FEM -.34346104
.04536453 -7.571 .0000 .11260504 UNION
.05752770 .01350031 4.261
.0000 .36398559 ED .11028379
.00510008 21.624 .0000 12.8453782
Constant 4.01913257 .07724830 52.029
.0000
25
Testing for Effects An LM Test
26
LM Tests
-------------------------------------------------
- Random Effects Model v(i,t) e(i,t) u(i)
Unbalanced Panel Estimates Vare
.216794D02 (T1) 1525
Varu .958560D01 (T2)
1079 Corrv(i,t),v(i,s)
.306592 (T3) 825 Lagrange
Multiplier Test vs. Model (3) 4419.33 (T4)
926 ( 1 df, prob value .000000)
(T5) 1051 (High values of LM
favor FEM/REM over CR model.) (T6) 1200
Baltagi-Li form of LM Statistic 1618.75
(T7) 887 ---------------------------------
----------------- ------------------------------
-------------------- Random Effects Model
v(i,t) e(i,t) u(i) Estimates Vare
.210257D02 Balanced Panel
Varu .860646D01
T 7 Corrv(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 Lhsdocvis Rhsone,hhninc,age,female,e
duc panel
27
Testing for Effects Moments
28
Testing (2)
29
Testing for Effects
? Obtain OLS residuals Regress
lhslwagerhsfixedx,varyingxrese ? Vector of
group sums of residuals Calc T 7 Groups
595 Matrix tebarTgxbr(e,person) ? Direct
computation of LM statistic Calc
listlmGroupsT/(2(T-1))
(tebar'tebar/sumsqdev - 1)2 ? Wooldridge chi
squared (N(0,1) squared) Create
e2ee Matrix e2iTgxbr(e2,person) Matrix
ridirp(tebar,tebar)-e2i sumriri'1 Calc
listz2(sumri)2/ri'ri
LM .37970675705025540D04 Z2
.16533465085356830D03
30
Two Way Random Effects Model
31
One Way REM
32
Two Way REM
Note sum .102705
33
Hausman Test for FE vs. RE
Estimator Random Effects EciXi 0 Fixed Effects EciXi ? 0
FGLS (Random Effects) Consistent and Efficient Inconsistent
LSDV (Fixed Effects) Consistent Inefficient Consistent Possibly Efficient
34
Hausman Test for Effects
ß does not contain the constant term in the
preceding.
35
Computing the Hausman Statistic
ß does not contain the constant term in the
preceding.
36
(No Transcript)
37
(No Transcript)
38
Hausman Test?
What went wrong? The matrix is not positive
definite. It has a negative characteristic
root. The matrix is indefinite. (Software such as
Stata and NLOGIT find this problem and refuse to
proceed.) Properly, the statistic cannot be
computed. The naïve calculation came out
positive by the luck of the draw.
39
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.

40
Variable Addition
41
Application Wu Test
NAMELIST XV exp,expsq,wks,occ,ind,south,smsa,
ms,union,ed,fem create expbgroupmean(exp,pds7
) create expsqbgroupmean(expsq,pds7) create
wksbgroupmean(wks,pds7) create
occbgroupmean(occ,pds7) create
indbgroupmean(ind,pds7) create
southbgroupmean(south,pds7) create
smsabgroupmean(smsa,pds7) create
unionbgroupmean(union,pds7) create msb
groupmean(ms,pds7) namelist xmeans
expb,expsqb,wksb,occb,indb,southb,smsab,msb,
unionb REGRESS Lhs lwage Rhs
xmeans,Xv,one panel random MATRIX bmean
b(19) vmean varb(19,19) MATRIX List
Wu bmean'ltvmeangtbmean
42
Means Added
43
Wu (Variable Addition) Test
44
Basing Wu Test on a Robust VC
? Robust Covariance matrix for REM Namelist
XWU wks,occ,ind,south,smsa,union,exp,expsq,ed,bl
k,fem, wksb,occb,indb,southb,smsab,
unionb,expb,expsqb,one Create ewu lwage -
xwu'b Matrix Robustvc ltXwu'XwugtGmmw(xwu,ewu
,_stratum)ltXwU'xWUgt Stat(b,RobustVc,Xwu)
Matrix Means b(1219)VmeansRobustVC(1219
,1219) List RobustWMeans'ltVmeansgtMean
s
45
Robust Standard Errors
46
Fixed vs. Random Effects
ß does not contain the constant term in the
preceding.
47
Another Comparison (Baltagi p. 20)
48
A Hierarchical Linear ModelInterpretation of the
FE Model
49
Hierarchical Linear Model as REM
-------------------------------------------------
- Random Effects Model v(i,t) e(i,t) u(i)
Estimates Vare
.235368D-01 Varu
.110254D00
Corrv(i,t),v(i,s) .824078
Sigma(u) 0.3303
-----------------------------------------------
--- -----------------------------------------
----------------------- Variable Coefficient
Standard Error b/St.Er.PZgtz Mean of
X -------------------------------------------
--------------------- OCC -.03908144
.01298962 -3.009 .0026 .51116447
SMSA -.03881553 .01645862 -2.358
.0184 .65378151 MS -.06557030
.01815465 -3.612 .0003 .81440576 EXP
.05737298 .00088467 64.852
.0000 19.8537815 FEM -.34715010
.04681514 -7.415 .0000 .11260504 ED
.11120152 .00525209 21.173 .0000
12.8453782 Constant 4.24669585
.07763394 54.702 .0000
50
Evolution Correlated Random Effects
51
Mundlaks Estimator
Mundlak, Y., On the Pooling of Time Series and
Cross Section Data, Econometrica, 46, 1978, pp.
69-85.
52
Correlated Random Effects
53
Mundlaks Approach for an FE Model with Time
Invariant Variables
54
Mundlak Form of FE Model
----------------------------------------------
------------------ Variable Coefficient
Standard Error b/St.Er.PZgtz Mean of
X -------------------------------------------
--------------------- x(i,t)

OCC -.02021384 .01375165 -1.470
.1416 .51116447 SMSA -.04250645
.01951727 -2.178 .0294 .65378151 MS
-.02946444 .01915264 -1.538
.1240 .81440576 EXP .09665711
.00119262 81.046 .0000 19.8537815 z(i)

FEM -.34322129
.05725632 -5.994 .0000 .11260504 ED
.05099781 .00575551 8.861 .0000
12.8453782 Means of x(i,t) and
constant
Constant 5.72655261 .10300460
55.595 .0000 OCCB -.10850252
.03635921 -2.984 .0028 .51116447 SMSAB
.22934020 .03282197 6.987 .0000
.65378151 MSB .20453332
.05329948 3.837 .0001 .81440576 EXPB
-.08988632 .00165025 -54.468 .0000
19.8537815 Variance Estimates
Vare
.0235632 Varu .0773825