Palgrave Handbook of Econometrics: Applied Econometrics

494 Discrete Choice Modeling

is an ambiguous virtue in this context. As we have seen, the raw coefficients are
of questionable value in interpreting the model – in order to translate them into
useful quantities we have computed partial effects and predicted probabilities. But
the semiparametric models specifically program around the assumption of a fixed
distribution and thus sacrifice the ability to compute partial effects or probabili-
ties. What remains is the estimator ofθand, in some cases, a covariance matrix
that can be used to test the significance of coefficients or to test hypotheses about
restrictions on structural coefficients.^12 Perhaps for these reasons, applied work in
binary choice remains overwhelmingly dominated by the parametric models.

11.3.5 Endogenous right-hand-side variables

The presence of endogenous right-hand-side variables in a binary choice model
presents familiar problems for estimation. The problem is made worse in nonlinear
models because, even if one has an instrumental variable readily at hand, it may
not be immediately clear what is to be done with it. The instrumental variable
estimator for the linear model is based on moments of the data, the variances and
covariances. In a binary choice setting, we are not using any form of least squares
to estimate the parameters, so the instrumental variable (IV) method would appear
not to apply. Generalized method of moments is a possibility. Consider the model:

di∗=x′iβ+γzi+εi

di= 1 (di∗> 0 )

E[εi|zi]=g(zi)=0.

Thusziis endogenous in this model. The MLEs considered earlier will not con-
sistently estimate (β,γ). (Without an additional specification that allows us to
formalize Prob(di = 1 |xi,zi), we cannot state what the MLE will, in fact, esti-
mate.) Suppose that we have a relevant (not “weak”) instrumental variable,wi,
such that:

E[εi|wi,xi]= 0

E[wizi] =0.

A natural instrumental variable estimator would be based on the “moment”
condition:

E

[

(d∗i−x′iβ−γzi)

(

xi

wi

)]

= 0.

However,d∗iis not observed:diis, but the “residual,”di−x′iβ−γzi, would have no
meaning even if the true parameters were known.^13 One approach that was used
in Avery, Hansen and Hotz (1983), Butler and Chatterjee (1997) and Bertschek and
Lechner (1998) is to assume that the instrumental variable is orthogonal to the
residual [di−#(x′iβ+γzi)], i.e.:

E

[

[di−#(x′iβ−γzi)]

(

xi

wi

)]

= 0.