Testing under Weak Identification with Conditional Moment Restrictions∗

We propose a semiparametric test for the value of coefficients in models with conditional moment restrictions which has correct size regardless of identification strength. The test is in essence an Anderson–Rubin (AR) test using nonparametrically estimated instruments to which we apply a standard error correction. We show that the test is (1) always size-correct, (2) consistent when identification is not too weak, and (3) asymptotically equivalent to an infeasible AR test when identification is sufficiently strong. We moreover prove that under homoskedasticity and strong identification our test has a limiting noncentral chi–square distribution under a sequence of local alternatives, where the noncentrality parameter is given by a quadratic form of the inverse of the semiparametric efficiency bound.