Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 499

For convenience, denote the “known”x′itβasbit. The first-order condition for
maximizing lnLwith respect toαi, given knownβ, is:

∂lnLi/∂bit=

∑Ti

t= 1

[dit−F(αi+bit)]=0.

This is one equation in one unknown that can be solved iteratively to provide an
estimate ofαi. The resulting estimator is inconsistent, sinceTiis fixed – the resulting
estimates are also likely to be highly variable because of the small sample sizes.
However, the inconsistency results not because it converges to something other
thanαi. The estimator is inconsistent because its variance is O(1/Ti). Consequently,
an estimator of the average partial effects:

ˆδ=^1

n

∑n

i= 1

∑Ti

t= 1 f(αˆi+x

′

itβˆ)βˆ,

may yet provide a useful estimate of the partial effects. This estimator remains to
be examined empirically or theoretically.
The fixed effects model has the attractive aspect that it is a robust specifica-
tion. The four shortcomings listed above, especially items 2 and 4, do reduce its
appeal, however. The wisdom behind the linear model does not carry over to binary
choice models because the estimation and inferential problems change substan-
tively in nonlinear settings. The statistical aspects of the random effects model
discussed next are more appealing. However, the model’s assumption of orthog-
onality between the unobserved heterogeneity and the included variables is also
unattractive. The Mundlak (1978) device is an intermediate step between these two
that is sometimes used. This approach relies on a projection of the effects on the
time invariant characteristics and group means of the time variables;

αi=z′iγ+π 0 +x′iπ+σuui, whereE[ui|xi]=0 and Var[ui|xi]=1.

(The location parameterπ 0 accommodates a non-zero mean while the scale
parameter,σu, picks up the variance of the effects, so the assumptions of zero
mean and unit variance foruiare just normalizations.) Inserting this into the fixed
effects model produces a type of random effects model:

dit∗=xit′β+z′iγ+π 0 +x′iπ+σuui+εit, t=1,...,Ti, i=1,...,n

dit=1ifd∗it>0, anddit=0 otherwise.

If the presence of the projection on the group means successfully picks up the corre-
lation betweenαiandxit, then the parameters (β,γ,π 0 ,π,σu)can be estimated by
maximum likelihood (ML) as a random effects model. The remaining assumptions
(functional form and distribution) are assumed to hold (at least approximately), so
that the random effects treatment is appropriate.

11.3.6.3 Random effects models and estimation

As suggested in the preceding section, the counterpart to a random effects model
for binary choice would be:

dit∗=xit′β+z′iγ+σuui+εit, t=1,...,Ti, i=1,...,n,