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Re: st: Reduced Form with biprobit


From   "Steven J. Carter" <sjcarter@uci.edu>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Reduced Form with biprobit
Date   Thu, 08 May 2008 10:39:18 -0700

Partha,

Yes, I only generated one data set, with 1000 observations. I will try to see what happens when I generate many data sets.

You confirmed my suspicions that the reduced form was incorrect.

Thanks again for your response.

Steven

Partha Deb wrote:

Steven,

From your description it seems that you've only generated one set of data. If that's the case, there is nothing in the theory that predicts your point estimate will be close to the truth. If you did your experiment many times, the average of the estimates should be close to the truth, esp. as the sample size gets larger.

Your reduced form is not correct - unless you want y2* on the RHS of the y1* equation (not y2). The fact that the estimate you get when you "back it out" is closer says nothing about its correctness.

Hope this helps.

Partha


Steven J. Carter wrote:

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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