Econometric Analysis of Cross Section and Panel Data - Free

Discrete Response Models 473
The model is applicable when y 2 is correlated with u 1 because of omitted variables
or measurement error. It can also be applied to the case where y 2 is determined jointly
with y 1 , but with a caveat. If y 1 appears on the right-hand side in a linear structural
equation for y 2 , then the reduced form for y 2 cannot be found with v 2 having the
stated properties. However, if y 1 appears in a linear structural equation for y 2, then
y 2 has the reduced form given by equation (15.40); see Maddala (1983, Chapter 7)
for further discussion.
The normalization that gives the parameters in equation (15.39) an average partial
e¤ect interpretation, at least in the omitted variable and simultaneity contexts, is
Varðu 1 Þ¼1, just as in a probit model with all explanatory variables exogenous. To
see this point, consider the outcome on y 1 at two di¤erent outcomes of y 2 , say y 2 and
y 2 þ 1. Holding the observed exogenous factors fixed at z 1 , and holding u 1 fixed, the
di¤erence in responses is
1½z 1 d 1 þ a 1 ðy 2 þ 1Þþu 1 b 0Š
1½z 1 d 1 þ a 1 y 2 þ u 1 b 0Š
(This di¤erence can take on the values 1, 0, and 1.) Because u 1 is unobserved, we
cannot estimate the di¤erence in responses for a given population unit. Nevertheless,
if we average across the distribution of u 1 , which is Normalð0; 1Þ, we obtain
F½z 1 d 1 þ a 1 ðy 2 þ 1ÞŠ
Fðz 1 d 1 þ a 1 y 2 Þ
Therefore, d 1 and a 1 are the parameters appearing in the APE. [Alternatively, if we
begin by allowing s1 2 ¼ Varðu 1Þ > 0 to be unrestricted, the APE would depend on
d 1 =s 1 and a 1 =s 1 , and so we should just rescale u 1 to have unit variance. The variance
and slope parameters are not separately identified, anyway.] The proper normalization
for Varðu 1 Þ should be kept in mind, as two-step procedures, which we cover in
the following paragraphs, only consistently estimate d 1 and a 1 up to scale; we have to
do a little more work to obtain estimates of the APE. If y 2 is a mismeasured variable,
we apparently cannot estimate the APE of interest: we would like to estimate the
change in the response probability due to a change in y2 , but, without further assumptions,
we can only estimate the e¤ect of changing y 2 .
The most useful two-step approach is due to Rivers and Vuong (1988), as it leads
to a simple test for endogeneity of y 2 . To derive the procedure, first note that, under
joint normality of ðu 1 ; v 2 Þ, with Varðu 1 Þ¼1, we can write
u 1 ¼ y 1 v 2 þ e 1
ð15:42Þ
where y 1 ¼ h 1 =t 2 2 , h 1 ¼ Covðv 2 ; u 1 Þ, t 2 2 ¼ Varðv 2Þ, and e 1 is independent of z and
v 2 (and therefore of y 2 ). Because of joint normality of ðu 1 ; v 2 Þ, e 1 is also normally