500 Discrete Choice Modeling
whereE[ui|xit]=0 and Var[ui|xit] = 1 and:
dit=1ifd∗it>0, anddit=0 otherwise.(Since the random effects model can accommodate time invariant characteristics,
we have reintroducedziinto the model.) The random effects model is fitted by ML
assuming normality forεitandui. (The most common application is the random
effects probit model.)
To begin, suppose the common effect is ignored, and the “pooled” model is
fitted by simple ML, ignoring the presence of the heterogeneity. The (incorrectly)
assumed model is:
Prob(dit= 1 |xit)=F(x′itβ+z′iγ).In the presence ofui, the correct model is:
Prob(dit= 1 |xit)=Prob(εit+σuui<x′itβ+z′iγ)=Prob⎛
⎜
⎝εit+σuui
√
1 +σu^2<
x′itβ+z′iγ
√
1 +σu^2⎞
⎟
⎠=Prob(
vit<x′itβu+z′iγu)
, vit∼N[0, 1].Thus the marginal probability thatditequals one obeys the assumptions of the
familiar probit. However, the coefficient vector is notβ, butβu=β/( 1 +σu^2 )^1 /^2 ,
and likewise forγ. The upshot is that ignoring the heterogeneity (random effect)
is not so benign here as in the linear regression model. In the regression case,
ignoring a random effect that is uncorrelated with the included variables produces
an inefficient, but consistent, estimator.
In spite of the preceding result, it has become common in the applied literature
to report “robust,” “cluster corrected” asymptotic covariance matrices for pooled
estimators such as the MLE above. The underlying justification is that, while the
MLE may be consistent (though it rarely is, as exemplified above), the asymptotic
covariance matrix should account for the correlation across observations within a
group. The corrected estimator is:
Est.Asy.Var[
θˆMLE]
=[∑
n
i= 1∑Ti
t= 1 Hit]− 1 [∑
n
i= 1(∑
Ti
t= 1 git)(∑
Ti
t= 1 g′
it)]×[∑
n
i= 1∑Ti
t= 1 Hit]− 1
,whereHit=∂^2 lnF(qit(x′itβ+z′iγ))/∂θ∂θ′andgit=∂lnF(qit(x′itβ+z′iγ))/∂θand all
terms are computed at the pooled MLE. The estimator has a passing resemblance
to the White (1980) covariance estimator for the least squares coefficient estimator.
However, the usefulness of this estimator rests on the assumption that the pooled
estimator is consistent, which will generally not be the case.