An Improved Inference for IV Regressions

Researchers often report empirical results that are based on low-dimensional IVs, such as the shift-share IV, together with many IVs. Could we combine these results in an efficient way and take advantage of the information from both sides? In this paper, we propose a combination inference procedure to solve the problem. Specifically, we consider a linear combination of three test statistics: a standard cluster-robust Wald statistic based on the low-dimensional IVs, a leave-one-cluster-out Lagrangian Multiplier (LM) statistic, and a leave-one-cluster-out Anderson-Rubin (AR) statistic. We first establish the joint asymptotic normality of the Wald, LM, and AR statistics and derive the corresponding limit experiment under local alternatives. Then, under the assumption that at least the low-dimensional IVs can strongly identify the parameter of interest, we derive the optimal combination test based on the three statistics and establish that our procedure leads to the uniformly most powerful (UMP) unbiased test among the class of tests considered. In particular, the efficiency gain from the combined test is of ``free lunch" in the sense that it is always at least as powerful as the test that is only based on the low-dimensional IVs or many IVs.