st: Re: 3SLS with different instruments for different equations

[Christopher Baum <kit.baum@bc.edu>]

<>
On Mar 15, 2011, at 2:33 AM, May wrote:

> 
> Y1 = b0 + b1*Y2 + b2*Y3 + b3*X1 + b4*X2
> 
> Y2 = k0 + k1*Y1 + k2*IV1 + k3*X1 + k4*X2
> 
> Y3 = j0 + j1*Y1 + j2*IV2 + j3*X1 + j4*X2
> 
> Y1 is endogenous to Y2 and Y3. But Y2 and Y3 are not endogenous.

This makes no sense. If Y2 and Y3 are on the LHS, and Y1 is on the RHS of that equation, all three Ys are endogenous. You can derive the reduced form for this model by solving the three equations, which will give you

Y1 = r1(x1, x2, iv1, iv2) + error
Y2 = r2(x1, x2, iv1, iv2) + error
Y3 = r3(x1, x2, iv1, iv2) + error

where the errors are combinations of the original three errors.

The equation for Y1 is exactly ID by the order condition as you have two endogenous regressors and two excluded instruments.

The equations for Y2 and Y3 are also exactly ID as each has one excluded regressor from the set of exogenous variables.

You cannot apply any additional exclusion restrictions on the instruments, as you have only just enough of them to ID each equation. 

All three Ys are obviously jointly determined in this system, as an analysis of the reduced form would show.

Kit

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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