Econometrics
Lecture Notes
Hayashi, Chapter 5d
Panel Data: Classical View
The Classical Model
• yi = Zid + ai + hi
yim = zim’d + ai + him
yim = fim’b + bi’g + ai + him
(i=1,…n; m=1,…M)
• E(hi) = 0, E(zimhim) = 0, E(aihim) = 0
• Note: E(zimai) = 0 is not assumed.
• E(hihi’) = sh2IM
Fixed-Effects Estimator
• First Difference Estimates
If m is a time index, then the model can be
transformed as:
yim = fim’b + him
(i=1,…,n; m=2,…,M)
Fixed-Effects Estimator
• Between Estimates
yi fi 'β bi' γ a i hi
1 M 1 M
where yi yim , fi M fim ,
M m 1 m 1
1 M sh
2
hi
M m 1
him , E(hi ) 0, E(hi hi' )
M
IM
Fixed-Effects Estimator
• Within Estimates
yim yi (fim fi ) β (him hi )
'
*
yim fimβ him
*' *
E(hi*) = 0, E(fim*him*’) = 0,
E(hi*hi*’) = sh2[IM-(1/M)1MxM] = sh2Q
Random-Effects Estimator
• Random-Effects Model
yim = zim’d + (ai + him)
eim = ai + him
• E(ei) = 0, E(zimeim) = 0
E(eiei’) = sa21MxM sh2IM = S
yim yi (z im zi )' δ [(1 )ai (him hi )]
y* z*' δ ε*
im im im
sh
2
where 1
Msa sh
2 2
Parameters Estimation
• To obtain the between estimates, OLS is used.
• To obtain the within estimates, pooled OLS is
used (with correction of the degrees of freedom).
• Random-effects model requires the estimate of
from the between and within estimates of
variances. Then pooled OLS is used on the
transformed model.
• Using xtreg in Stata.
Hypotheses Testing
• Testing for fixed-effects
– F-test for ai = 0 jointly
• Testing for random-effects
– Breusch-Pagan LM c2-test for sa2 = 0
2
n M 2
e im
ˆ
nM i 1 m 1
1 c 2 (1)
2( M 1) n M ˆ 2
e im
i 1 m 1
R 2
• Three concepts of R2
– Between yi fi 'β bi' γ ai hi
– Within (from pooled OLS)
yim yi (fim fi )' β (him hi )
yim yi (z im zi )' δ [(1 )ai (him hi )]
– Total
yim = fim’b + bi’g + ai + him
(i=1,…n; m=1,…M)
Parameters Estimation
• Random-effects model can also be estimated with
maximum likelihood method.
• GLS estimation (with non-classical variance-
covariance matrix, see xtgls in Stata)
• Mixed fixed-effects and random-effects (xtmixed
in Stata)
• Autocorrelation in panels (xtregar in Stata)
• Dynamic panel data models (xtabond in Stata)
Endogenous Regressors
• yi = Zid + ai + hi
• yi = Z1id1 + Z2id2 + ai + hi
– Z1i is predetermined, Z2i is endogenous.
– Xi = [X1i,X2i] is exogenous.
X1i is included, X2i is excluded instruments.
Set X1i = Z1i, #X2i #Z2i
Endogenous Regressors
• yi = Zid + ai + hi
with instrumental variables Xi.
E(Ximhim)=0, E(Zimhim)0, E(aihim)=0
E(hi)=0, E(hihi’)=sh2IM
• IV method (xtivreg in Stata) can be applied to:
– First-Difference Estimates
– Between Estimates
– Fixed-Effects or Within Estimates
– Random-Effects Estimates
Conditional Heteroscedasticity
• Robust (or White) estimates of standard errors can
be obtained to improve consistency of the
estimates.
• GMM methods can be applied to panel data
models with conditional heteroscedasticity and
random regressors (xtivreg2 in Stata).
• Testing for overidentifying restrictions such as
Hansen or Sargan test for panel data models is
possible (xtoverid in Stata).