st: re: ivreg2: No validity tests if just-identified?

[Christopher Baum <baum@bc.edu>]

<>
John said

Out of interest, if one could specify how the error terms are handled,  
then it is possible to test for over-identifying restrictions,  
correct? That is:
y = b0 + b1x_hat + e1
x = b11 + b12z + e12
The covariance between e1 and e12 is estimated in ivreg, right? Hence  
the model is just-identified. Constraining the covariance to be  
orthogonal would provide for an overidentifying test. However,  
theoretically, estimating this covariance is necessary to account for  
the common cause of x and y not included in the model (so it would be  
an unreasonable restriction to make, unless the model is perfect).  
Right?


As written, this is a recursive system (if we assume that the y  
equation contains x rather than 'xhat', whatever that may be). If the  
structural equation for y contains x, x is a stochastic linear  
function of z. If the errors on those two equations are distributed  
independently, there would be no problem with estimating the y  
equation with OLS. After all, what is exogenous to the y equation may  
well have some equation determining it.

The more common setup for an IV problem would be to write y = f(x) and  
x = g(y, z), so that these are simultaneous equations. Then you have  
an endogeneity problem for each equation, and even if their errors are  
independently distributed, there is a correlation between regressor  
and error. You could estimate the y equation with IV, as it would be  
exactly ID using z. You could not estimate the x equation, as it would  
be unidentified by the order condition.

I don't know how to constrain a covariance to be orthogonal; I presume  
what is meant is to constrain e1 and e12 to be orthogonal. But in the  
model as written, that would merely guarantee that OLS would be  
consistent.

Although you cannot carry out a test of overid restrictions on an  
exactly ID equation, you can test whether IV methods are required for  
consistency (see Baum-Schaffer-Stillman, Stata Journal 7:4, 2007,  
preprint available below):

use http://fmwww.bc.edu/ec-p/data/hayashi/griliches76.dta
ivreg2 lw s expr (iq=med), endog(iq)


Kit Baum   |   Boston College Economics and 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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