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st: Reduced Form with biprobit


From   "Steven J. Carter" <sjcarter@uci.edu>
To   statalist@hsphsun2.harvard.edu
Subject   st: Reduced Form with biprobit
Date   Thu, 08 May 2008 10:09:13 -0700

Dear Statalist,

I have a probit model with an endogenous dummy variable and I found some helpful hints on this thread (http://www.statacorp.com/statalist/archive/2007-05/msg00366.html.). I generated a data set to see if biprobit recovers the parameters, and it doesn't. However, when I model the reduced form, (substituting the rhs variables in equation 2 for y2 in equation 1), and back out the structural parameters, it is pretty close. So my question is, why does it work for the reduced form parameters, but not the recursive form? Also, wouldn't the reduced form be similar to the forbidden regression that is discussed in Wooldridge's Cross section and Panel data text for nonlinear models?

Here is my do file
********* Confusing bivariate probit question ******

* Note: I generated the data in another program because I am not quite familiar with stata statistics codes yet.

* this is a simulated bivariate probit dataset with an endogenous regressor
* model specified as
* y1*=g*y2+x*b1+e1, x includes constant in both equations
* y2*=x*b2+d*z+e2, where e1, e2 are normal with zero means, unit variances and correlation p
* true values for parameters are g=0, b1=[-.5 .07]', b2=[0 -.1]', d=.7, p=.5

*************
* do a bivariate probit as shown in greene (2003) pg 715-716
biprobit (y1= y2 x) (y2=x z)

* notice that the coefficients don't match the true values

* Now try a bivariate probit using reduced form:
* y1*=g*(b2*x+d*z)+b1*x
* =x*(g*b2+b1)+z*(g*d)
* =x*w1+z*w2
* y2*=b2*x+d*z+e2

biprobit (y1=x z) (y2=x z)
matrix define coef2=e(b)

* true reduced form parameters
matrix define rftrue=(.07, 0, -.5, -.1, .7, 0, .5)
matrix list coef2
matrix list rftrue
* though estimating the reduced form parameters, w1 and w2 in the first equation, solving for the structual parameters (g, b1) yields results closer * to the true parameters.

********* End confusing question**************

I appreciate any input on this.

Best

Steven

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