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st: endogenous regression with OLS first stage and logit main regression


From   Karen Ruckman <ruckman@sfu.ca>
To   statalist@hsphsun2.harvard.edu
Subject   st: endogenous regression with OLS first stage and logit main regression
Date   Fri, 21 Oct 2011 10:05:50 -0700 (PDT)

Hello,

I am trying to run an instrumental variables regression with an OLS first stage and logit (or probit if necessary) in the main regression.  We have a system that is "triangular" which means that our endogenous regressor isn't a function of the dependent variable.  Therefore, our system looks like this:

main regression (logit or probit):
y1 = y2 + x1 + e

first stage (OLS):
y2 = instrument + error

The problem with using IVPROBIT is that the first stage will use all the exogenous regressors from the main regression as instruments, which we don't want or need as it won't bias the results to use only our one instrument.  There is a very useful discussion about how to use our one instrument in the first stage instead of all the exogenous regressors at the bottom of the FAQ below:
   http://www.stata.com/support/faqs/stat/ivreg.html
It says to simply regress the first stage and substitute y2hat for y2 into the main regression.  No problems there.  But we run into trouble when trying to correct the variance-covariance matrix with the "correct mean square error".  The code is written for a system where the first and main stages are both OLS.   Is there a mean square error for logit?  STATA doesn't produce it.  

Here is the code which recalculates the var-covar matrix after running an OLS main stage regression with y2hat substituted in.  
 . scalar realmse = r(mean)*r(N)/e(df_r) 
 . matrix bmatrix = e(b)
 . matrix Vmatrix = e(V)
 . matrix Vmatrix = e(V) * realmse / e(rmse)^2
 . ereturn post bmatrix Vmatrix, noclear
 . ereturn display

Any help with the issue with correcting the var-covar matrix would be most appreciated.  

Thanks!!

Karen Ruckman
Associate Professor
Beedie School of Business, SFU
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