st: endogenous multivariate probit

["Andrea Morescalchi" <morescalchi@ec.unipi.it>]

Dear all,
I am trying to estimate a probit equation with an 
endogenous discrete variable. Whenever the discrete 
endogenous variable is just binary, one should use 
-biprobit- as suggested in Greene 2003 and as was 
suggested also in a previous thread.
The model would look like this:
y = x1-xn end
end = x1-xn instr
where y and end are dummies and instr is an instrument for 
end.

In my case, the discrete variable end is not binary but 
has 4 categories, not ordered.
One way to estimate this model may be to use a two-steps 
procedure as suggested by Rivers-Vuong (1988) and 
explained in Wooldridge (15.7.2). In the first step I 
should run a multinomial logit for end (I have 3 
instruments) and then plug in the residuals for three 
dummies out of four in the main equation (a valid test for 
endogeneity is simply to look a the t-statistic of the 
three dummies). Since beta coefficients of the probit 
function are estimated up to a scale in the 2nd step, one 
should be aware of this when computing APEs. Unfortunately 
my references cover this issue only for cases when the 
endogenous variable is binary, so I am not sure about the 
proper rescaling I should do to compute APEs.

So my question is: may I go for a multivariate probit to 
estimate this model (Jenkins' -mvprobit- package)?
Basically, instead of treating the endogenous variable 
with an mlogit I would create three more probit equations. 
The full model would look like this:
y = x1-xn end1 end2 end3
end1 = x1-xn instr1 instr2 instr3
end2 = x1-xn instr1 instr2 instr3
end3 = x1-xn instr1 instr2 instr3

Thanks!
Of course, also comments on how to implement the two-steps 
approach would be very useful.

Andrea
*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/