“Avoiding Invalid Instruments and Coping with Weak Instruments”
Here are some notes on “Avoiding Invalid Instruments and Coping with Weak Instruments” by Michael P. Murray, Journal of Economic Perspectives, Volume: 20 | Issue: 4 Fall 2006 111-132
1. 2SLS uses only that part of the variation in X that appears as variation in the fitted values, the Xhat. With less variation, there is less information, so efficiency is lost.
2. The Hausman Test helps you know when you can use OLS instead of 2SLS. If they are close, then you can feel safer in using OLS.
3. Suppose you know that Z1 is a good instrument, but not Z2. You regress X on Z1 and Z2 to get Xhat. If you regress Y on Xhat and Z2 and find Z2 has a significant coefficient, you can reject Z2 as an instrument– it is correlated with Y.
4. The Sargan (1958) test tests whether ANY of the instruments are invalid, under the assumption that at least ONE is valid. It regresses the residuals from a regression of Y on Xhat on Z1 and Z2. If the R2 from that residual regression is big enough, we can deduce that either Z1 or Z2 is invalid.
5. The Sargan Test is an R2-based test, so it isn’t useful for nonlinear regressions, where R2 doesn’t have meaning. Hansen’s J- test is the nonlinear, GMM, analog. It tests whether any of the moment conditions are invalid.
6. When we use Z1 to instrument for X, if Z1 is only weakly correlated with X then our 2SLS standard errors are biased down. (A claim on p. 122– I don’t know why.)
7. For a special case, Hahn and Hausman (2005) show that the bias of 2SLS is bigger than that of OLS if the number of instruments is greater than (sample size)(R2 of the regression of Y on Xhat).
8. Stock and Yogo (2005) have a test for the null hypothesis that the bias of 2SLS is less than 10% the bias of OLS. When there is just one X instrumented, it uses an F-statistic of the null that the instruments Z1 and Z2 have zero coefficients when X is regressed on them in a 1st-stage regression.
9. There is something called a “conditional likelihood ratio test” which can test for weak instruments when there is only one troublesome X.