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

From   Christopher Baum <>
Subject   st: re: ivreg2: No validity tests if just-identified?
Date   Fri, 17 Apr 2009 14:54:21 -0400

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):

ivreg2 lw s expr (iq=med), endog(iq)

Kit Baum   |   Boston College Economics and DIW Berlin   |
An Introduction to Stata Programming   |
An Introduction to Modern Econometrics Using Stata   |

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