Earlier today, Karen Conway <ksconway@cisunix.unh.edu> asked:
> Just a follow-up question to Marc's clear discussion of bivariate
> probit... Suppose that one finds rho is not significantly different
> from zero in the case of a simultaneous probit model. Does that mean
> that you can treat the endogenous dichotomous regressor as exogenous
> when you estimate the two probits separately? In other words, is rho
> a de facto exogeneity test as well? My understanding is that you can
> include the observed dichotomous regressor on the rhs, even though its
> endogenous, BECAUSE you are estimating with a bivariate probit
> (Greene 2003, p. 716).
The short answer is yes, rho can be interpreted as a test of exogeneity
here.
In the recursive simultaneous equations probit model, we have
y1 = X1'b1 + e1
y2 = X2'b2 + a*y1 + e2
where y1 and y2 are dichotomous and e1 and e2 are bivariate normal.
We can "ignore" the endogeneity and use -biprobit- because both y1
and y2 are dichotomous and because we assume bivariate normality for
the errors.
If rho is zero, then random shocks to the second equation (e2) have
no effect on the outcome y1; and, hence, y1 is exogenous.
Hope this helps.
-- Brian
-- bpoi@stata.com
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