Testing with many weak instruments

Abstract

Abstract This paper establishes the asymptotic distributions of the likelihood ratio (LR), Anderson–Rubin (AR), and Lagrange multiplier (LM) test statistics under “many weak IV asymptotics.” These asymptotics are relevant when the number of IVs is large and the coefficients on the IVs are relatively small. The asymptotic results hold under the null and under suitable alternatives. Hence, power comparisons can be made. Provided k 3 / n → 0 as n → ∞ , where n is the sample size and k is the number of instruments, these tests have correct asymptotic size. This holds no matter how weak the instruments are. Hence, the tests are robust to the strength of the instruments. The asymptotic power results show that the conditional LR test is more powerful asymptotically than the AR and LM tests under many weak IV asymptotics.