This paper studies the asymptotic validity of the
Anderson–Rubin (AR) test and the
J test for overidentifying
restrictions in linear models with many instruments.
When the number of instruments increases at the same
rate as the sample size, we establish that the
conventional AR and
J tests are asymptotically
incorrect. Some versions of these tests, which are
developed for situations with moderately many
instruments, are also shown to be asymptotically
invalid in this framework. We propose modifications
of the AR and J
tests that deliver asymptotically correct sizes.
Importantly, the corrected tests are robust to the
numerosity of the moment conditions in the sense
that they are valid for both few and many
instruments. The simulation results illustrate the
excellent properties of the proposed tests.