Efficient Inference in the Classical IV Regression Model with Weak Identification: Asymptotic Power Against Arbitrarily Large Deviations from the Null Hypothesis

This paper considers efficient inference for the coefficient on the endogenous variable in linear regression models with weak instrumental variables (Weak-IV) and homoskedastic errors. We focus on the alternative hypothesis determined by an arbitrarily large deviation from the null hypothesis. The efficient rotation-invariant and asymptotically similar test turns out to be infeasible as it depends on the unknown correlation between structural and first-stage errors (the degree of endogeneity). We compare the asymptotic power properties of popular Weak-IV-robust tests, focusing on the Anderson-Rubin (AR) and the Conditional Likelihood Ratio (CLR) tests. We find that their relative power performance depends on the degree of endogeneity in the model and the number of IVs. Unexpectedly, the AR test outperforms the CLR when the degree of endogeneity is small and the number of IVs is large. We also describe a test that is optimal when IVs are strong and, when IVs are weak, has the same asymptotic power as the AR test against arbitrarily large deviations from the null.