# Re: st: IV estimation for probit models with binary endogenous variable...?

 From Antoine Terracol To statalist@hsphsun2.harvard.edu Subject Re: st: IV estimation for probit models with binary endogenous variable...? Date Thu, 02 Apr 2009 14:48:04 +0200

```I forgot to include this part of -cmp-'s help file in my previous message:

```
So some Stata estimation commands have wider applicability than many realize. sureg (X=Y) (Y=Z), isure typically matches ivreg X (Y=Z) exactly even though the documentation does not describe sureg as an instrumental variables (IV) estimator. (Iterated SUR is not a true ML estimator, but it converges to the same solution as ML-based SUR, as implemented, for example, in the demonstration command mysureg. See Pagan (1979) on the LIML-iterated SUR connection.) And biprobit (X=Y) (Y=Z) will consistently
```    estimate an IV model in which X and Y are binary.

Antoine

Antoine Terracol wrote:
```
Kit, I miht have misunderstood your comment, but I read in -cmp-'s help file that:
```
```
As a matter of algorithm, cmp is an SUR (seemingly unrelated regressions) estimator. It treats the equations as independent from each other except for modeling their underlying errors as jointly normally distributed. Mathematically, the likelihood it computes is conditioned on observing all right-side variables, including those that also appear on the left side of equations. However, it can actually fit a much larger class of models. Maximum likelihood (ML) SUR estimators, including cmp, are appropriate for an important class of simultaneous equation models, in which endogenous variables appear on the right side of structural equations as well as the left. Models of this kind for which ML SUR is
```    nevertheless consistent must satisfy two criteria:

```
1) They are recursive. In other words, the equations can be arranged so that the matrix of coefficients of the dependent variables in each others' equations is triangular. As emphasized above, this means the models have clearly defined stages, though there can be more than one equation
```        per stage.

```
2) Dependent variables in one stage enter subsequent stages only as observed. Returning to the example in the first paragraph, if C is a categorical variable modeled as ordered probit, then C, not the latent variable underlying it, call it C*, must figure in the model for D.
```

In the following example, -cmp- and -biprobit- give the same results

/*---------------------------*/
clear
set obs 1000
drawnorm e1 e2, cov(1,0.5\0.5,1)
drawnorm x1 x2 z1 z2
g endog=(1+z1+z2+e1>0)
g y=(1+endog+x1+x2+e2>0)
cmp (y = endog x1 x2) (endog  = z1 z2), ind(4 4)
biprobit (y = endog x1 x2) (endog  = z1 z2)
/*---------------------------*/

Antoine

Kit Baum wrote:
```
```<>   Antoine said

If you are ready to assume joint normality, then -biprobit- should do
the trick:

/*------------------------------*/
clear
set obs 10000
set seed 987654321
drawnorm e1 e2, cov(1,0.5\0.5,1)
drawnorm x1 x2 z1 z2
g endog=(1+z1+z2+e1>0)
g y=(1+endog+x1+x2+e2>0)
probit y endog x1 x2 /*biased*/
biprobit (y= endog x1 x2) (endog=z1 z2)

```
I'm not so sure. Stata will allow you to estimate that model, but it calls it the "seemingly unrelated bivariate probit" model. That model is described in Greene, Econometric Analysis 6ed (p. 817), as analogous to SUR ("in the same spirit as the seemingly unrelated regression model"): that is, two equations in which there are nothing but exogenous explanatory variables. The way in which Antoine has written the model is one in which you surely could use cmp, as it is recursive (y depends on endog, but endog does not depend on y). But I'm not sure that the assumptions of the SUBP model are satisfied here.
```
```
Greene (p. 817) describes a model in which an endogenous regressor is binary as a 'treatment effects' model and suggests that it should be treated as a selection problem.
```
```
Kit Baum | Boston College Economics & DIW Berlin | http://ideas.repec.org/e/pba1.html An Introduction to Stata Programming | http://www.stata-press.com/books/isp.html An Introduction to Modern Econometrics Using Stata | http://www.stata-press.com/books/imeus.html
```

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```
```

```
```

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```