On the Limiting and Empirical Distributions of IV Estimators When Some of the Instruments are Actually Endogenous

IV estimation is examined when some instruments may be invalid. This is relevant because the initial just-identifying orthogonality conditions are untestable, whereas their validity is required when testing the orthogonality of additional instruments by so-called overidentification restriction tests. Moreover, these tests have limited power when samples are small, especially when instruments are weak. Distinguishing between conditional and unconditional settings, we analyze the limiting distribution of inconsistent IV and examine normal first-order asymptotic approximations to its density in finite samples. For simple classes of models we compare these approximations with their simulated empirical counterparts over almost the full parameter space. The latter is expressed in measures for: model fit, simultaneity, instrument invalidity, and instrument weakness. Our major findings are that for the accuracy of large sample asymptotic approximations instrument weakness is much more detrimental than instrument invalidity. Also, IV estimators obtained from strong but possibly invalid instruments are usually much closer to the true parameter values than those obtained from valid but weak instruments.