IV estimation with valid and invalid instruments

While 2SLS is the most widely used estimator for simultaneous equation models, OLS may do better in finite samples. Here we demonstrate analytically that for the widely used simultaneous equation model with one jointly endogenous variable and valid instruments, 2SLS has smaller MSE error, up to second order, than OLS unless the R2 , or the F statistic of the reduced form equation is extremely low. We then consider the relative estimators when the instruments are invalid, i.e. the instruments are correlated with the stochastic disturbance. Here, both 2SLS and OLS are biased in finite samples and inconsistent. We investigate conditions under which the approximate finite sample bias or the MSE of 2SLS is smaller than the corresponding statistics for the OLS estimator. We again find that 2SLS does better than OLS under a wide range of conditions. We then present a method of sensitivity analysis, which calculates the maximal asymptotic bias of 2SLS under small violations of the exclusion restrictions. For a given correlation between invalid instruments and the error term, we derive the maximal asymptotic bias. We apply our results to IV estimation of the returns to education. We derive the bias in the estimated standard errors of 2SLS for the first time. This derivation also has implications for the test of over-identifying restrictions. Keywords: Instrumental Variables, 2SLS, Weak Instruments, Returns to Education. JEL Classification: C1, C3.