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 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 power results show that the conditional LR test is more powerful asymptotically
than the AR and LM tests under many weak IV asymptotics.