Estimation with Valid and Invalid Instruments

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 also consider the relative bias of 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, which we characterize as functions of observable statistics and one unobservable statistic.