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st: endogenous multivariate probit


From   "Andrea Morescalchi" <[email protected]>
To   [email protected]
Subject   st: endogenous multivariate probit
Date   Sun, 13 Feb 2011 15:23:21 +0100

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
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