Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression

This paper considers tests of the parameter on endogenous variables in an instru-
mental variables regression model. The focus is on determining tests that have some
optimal power properties. We start by considering a model with normally distrib-
uted errors and known error covariance matrix. We consider tests that are similar and
satisfy a natural rotational invariance condition. We determine a two-sided power
envelope for invariant similar tests. This allows us to assess and compare the power
properties of tests such as the conditional likelihood ratio (CLR), Lagrange multi-
plier, and Anderson-Rubin tests. We find that the CLR test is quite close to being
uniformly most powerful invariant among a class of two-sided tests.
The finite sample results of the paper are extended to the case of unknown error
covariance matrix and possibly non-normal errors via weak instrument asymptotics.
Strong instrument asymptotic results also are provided because we seek tests that
perform well under both weak and strong instruments.