Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 501

Efficiency is a moot point for this estimator, since the probit MLE estimatesβ
with a bias toward zero:

plimβˆMLE=βu

=β/( 1 +σu^2 )^1 /^2

=β( 1 −ρ^2 )^1 /^2 ,

whereρ^2 =Corr^2 [εit+ui,εis+ui] fort=s. Wooldridge (2002a) suggests that this
may not be an issue here, since the real interest is in the partial effects, which are,
for the correct model:

δit=∂Prob[dit= 1 |xit,zi]/∂xit=βuφ(x′itβu+z′iγu).

These would then be averaged over the individuals in the sample. It follows, then,
that the “pooled” estimator, which ignores the heterogeneity, does not estimate
the structural parameters of the model correctly, but it does produce an appropriate
estimator of the average partial effects.
In the random effects model, the observations are not statistically independent

• because of the commonui, the observations (di 1 ,...,di,Ti,ui) constitute aTi+ 1
  variate random vector. The contribution of observationito the log-likelihood is
  the joint density:

f(di 1 ,...,di,Ti,ui|Xi)=f(di 1 ,...,di,Ti|Xi,zi,ui)f(ui).

Conditioned onui, theTirandom outcomes,di 1 ,...,di,Ti, are independent. This
implies that (with the normality assumption now incorporated in the model) the
contribution to the log-likelihood is:

lnLi=ln

{[∏

Ti

t= 1 #

(

qit(x′itβ+z′iγ+σuui)

)]

φ(ui)

}

,

whereφ(ui)is the standard normal density. This joint density contains the unob-
servedui, which must be integrated out of the function to obtain the appropriate
log-likelihood function in terms of the observed data. Combining all terms, we
have the log-likelihood for the observed sample:

lnL=

∑n

i= 1 ln

[∫∞

−∞

(∏

Ti

t= 1 #

(

qit(x′itβ+z′iγ+σuui)

))

φ(ui)dui

]

. (11.7)

Maximization of the log-likelihood with respect to (β,σu) requires evaluation
of the integrals in equation (11.7). Since these do not exist in closed form,
some method of approximation must be used. The most common approach is
the Hermite quadrature method suggested by Butler and Moffitt (1982). The
approximation is written:
∫∞

−∞

(∏

Ti

t= 1 #

(

qit(x′itβ+z′iγ+σuui)

))

φ(ui)dui≈

1

√

π

∑H

h= 1 wh

∏Ti

t= 1

#

(

qit(x′itβ+z′iγ+

√

2 σuzh)

)

,