CHAPTER 13 Models for Panel Data 311
Notethatthemvariablesareexogenoustimeinvariantvariables,z1i andtheexogenous
timevaryingvariables,eithercondensedintothesinglegroupmeanorintherawform, with the full set of T
observations. To construct the estimator, we
the simple heteroscedastic model is not general enough for these data. If the null hypothesis is that the
disturbances are both homoscedastic and uncorrelated across groups, then these two tests are
inappropriate. A likelihood ratio test can be constructe
rms. To investigate this proposition before tting an extended model, we can use the
testsforhomoscedasticitysuggestedearlier.BasedontheOLSresults,theLMstatistic equals 46.63. The critical
value from the chi-squared distribution with four degrees of freedo
Table 13.4 also presents estimates for the groupwise heteroscedasticity model and for the full model
with cross-sectional correlation, with the corrections for rst-order autocorrelation. The lower part of
the table displays the recomputed group specic var
specic dummy variables sum to 1, so there are some redundant coefcients. The discussion in Section
13.3.3 shows that one way to remove the redundancy is to include an overall constant and drop one of
the time specic and one of the timedummy variables. The
Note that Wi is assumed to be, a T(1+K1 +K2 +L1 +L2) matrix. Since there is a
laggeddependentvariableinthemodel,itmustbeassumedthatthereareactuallyT+1 observationsavailableon
yit.Toavoidacumbersome,clutterednotation,wewillleave this distinction embedded i
SUMMARY AND CONCLUSIONS
The preceding has shown a few of the extensions of the classical model that can be obtained when panel
data are available. In principle, any of the models we have examined before this chapter and all those we
will consider later, i
Any vector whose elements sum to zero is a solution. There are T1 such independentvectors,so
T1characteristicrootsare (1) =0or =1.Premultiply theexpressionbyi
toobtaintheremainingcharacteristicroot.(Remembertoadd one to the result.) Now, collect terms to
CHAPTER 13 Models for Panel Data 333
[An interesting problem arises at this point. If one computes these autocorrelations using the standard
formula, then the results can be substantially affected because the group-specic residuals may not have
mean zero.
estimatorsdependonanincreaseinT,sotheyaregenerallynotwellsuitedtothetypes of data sets described in
Sections 13.213.8. Beck et al. (1993) suggest several problems that might arise when using this model
in small samples. If T <n, then with or without a cor
. If the equations are actually unrelatedthat is, if ij =0 for i
= jthen there is obviously no payoff to
GLS estimation of the full set of equations. Indeed, full GLS is equation by equation OLS.6 2. If the
equations have identical explanatory variablesth
where i indexes rms and t indexes years. Different restrictions on the parameters and the variances and
covariances of the disturbances will imply different forms of the model. By pooling all 100 observations
and estimating the coefcients by ordinary leas
. Compute the two separate OLS estimates of , their sampling variances, the estimates of 2 1 and 2 2,
and the R2s in the two regressions. b. Carry out the Lagrange multiplier test of the hypothesis that 2 1
= 2 2. c. Compute the two-step FGLS estimate of
Theinstrumentalvariablesetscontainedinvit whichhavebeensuggestedmightinclude the following from
within the model: xit and xi,t1 (i.e., current and one lag of all the time varying variables) xi1,.,xiT
(i.e.,allcurrent,pastandfuturevaluesofallthetimevarying
There are many settings in which the models of the previous chapters apply to a group of related
variables. In these contexts, it makes sense to consider the several models jointly. Some examples follow.
1. The capital asset pricing model of nance species
Poolthedataandcomputetheleastsquaresregressioncoefcientsofthemodel yit = +xit +it. b. Estimate
the xed effects model of (13-2), and then test the hypothesis that the constant term is the same for all
three rms. c. Estimate the random effects model of (13-
is moderately large and all estimators are consistent, so this result is to be expected.) We shall examine
the effect of assuming that all ve rms have the same slope parameters in Section 14.2.3. For now, we
note that one of the effects is to inate the di
all, however.2 Section 14.2 below examines the general model in which each equation has its own xed
set of parameters, and examines efcient estimation techniques. Production and consumer demand
models are a special case of the general model in which the
e
There are a few results for unequal numbers of observations, such as Schmidt (1977), Baltagi, Garvin,
and Kerman(1989),Conniffe(1985),Hwang,(1990)andIm(1994).Butgenerally,thecaseofxed T isthenorm in
practice. 4ThisisthetestofAggregationBiasthatisthesubjec
The case of identical regressors is quite common, notably in the capital asset pricing model in empirical
nancesee Section 14.2.5. In this special case, generalized least squares is equivalent to equation by
equation ordinary least squares. Impose the ass
vector to vary across groups. The covariance structures model is, therefore, a testable special case.4 It
will be convenient in the discussion below to have a term for the particular kind of model in which the
data matrices are group specic data sets on t
regression analysis is F[J, MTK]= (R q)[R(X 1X)1R]1(R q)/J 1 /(MTK)
. (14-12)
The computation requires the unknown . If we insert the FGLS estimate based on (149)andusetheresultthatthedenominatorconvergestoone,then,inlargesamples, the statistic will behav
anticipated prot and replacement of the capital stock. We will now specify Iit = 1i +2iFit +3iCit +it.
Whether the parameter vector should be the same for all rms is a question that we shall study in this
chapter. But the disturbances in the investment eq
OLS and GLS will be the same if the K columns of X are a linear combination of exactly K characteristic
vectorsof .ByshowingtheequalityofOLSandGLShere,wehaveveriedtheconditionsofthecorollary. The
general result is pursued in the exercises. The intriguing
overalltisanaggregateofmediocretsforChryslerandWestinghouseandobviously
terribletsforGM,GE,andU.S.Steel.Indeed,theconventionalmeasureforGEbased on the same FGLS
residuals, 1eGEeGE/yGEM0yGE is16.7! We might use (14-11) to compare the t of the unrestricted
analyzedtheNerlovedataandaugmentedthedatasetwithcostsharedatatoestimate the complete demand
system. Appendix Table F14.2 lists Nerloves 145 observations with Christensen and Greenes cost share
data. Cost is the total cost of generation in millionsofdollar
To reiterate, the important result we have here is that in the SUR model, when all equations have the
same regressors, the efcient estimator is single-equation ordinary least squares; OLS is the same as GLS.
Also, the asymptotic covariance matrix of for t
E[|X1,X2,.,XM] = 0, E[|X1,X2,.,XM] = .
WeassumethatatotalofTobservationsareusedinestimatingtheparametersofthe M equations.3 Each
equation involves Km regressors, for a total of K =n i=1 Ki. We willrequire T >
Ki.Thedataareassumedtobewellbehaved,asdescribe
The consistency of sij follows from that of bi and bj. A degrees of freedom correction in the divisor is
occasionally suggested. Two possibilities are
s ij =
e iej [(TKi)(TKj)]1/2
and s ij =
e iej Tmax(Ki, Kj)
.12
The second is unbiased only if i equals j