Testing with many weak instruments

Provided k3/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.