A Uniformly Valid Test for Instrument Exogeneity

This paper studies the limiting behavior of the test for instrument exogeneity in linear models when there is uncertainty about the strength of the identification signal. We consider the test for conditional moment restrictions with an expanding set of constructed instruments. We establish the uniform validity of the standard normal asymptotic approximation, under the null, of this specification test over all possible degrees of model identification. As a result, this allows the researcher to use standard inference for testing instrument exogeneity without the need of any prior knowledge if the instruments are strong, semi-strong, weak, or completely irrelevant. Furthermore, we show that the test is consistent regardless of the instrument strength; i.e., even in cases (weak and completely irrelevant instruments) where the standard tests fail to exhibit asymptotic power. To obtain these results, we characterize the rate of the estimator under a drifting sequence for the identification signal. We illustrate the appealing properties of the test in simulations and an empirical application.