Valid Confidence Regions and Inference in the Presence of Weak Instruments

We investigate confidence intervals and inference for the instrumental variables model with weak instruments. Wald-based confidence intervals perform poorly in that the probability they reject the null is far greater than their nominal size. In the worst case, Wald-based confidence intervals always exclude the true paremeter value. Confidence intervals based on the LM, LR, and Anderson-Rubin statistics perform far better than the Wald. The Anderson-Rubin statistic always has the correct size, but LM and LR statistics have somewhat greater power. Performance of the LM and LR statistics is improved by a degrees-of-freedom correction in the overidentified ccase. We show that the practice of "pre- testing" by looking at the significance of the first -stage regression leads to extremely poor results when the instruments are very weak.