On the size of distortion of tests after an overidentifying restrictions pretest

In the linear instrumental variables model, we provide theoretical and Monte Carlo evidence for the size distortion of a two-stage hypothesis test that uses a test of overidentifying restrictions (OR) in the first stage. We derive a lower bound for the asymptotic size of the two-stage test. The lower bound is given by the asymptotic size of a test that rejects the null hypothesis when two conditions are met: the test of OR used in the first stage does not reject and the test in the second stage rejects. This lower bound can be as large as 1 — ε P , where ε P is the pretest nominal size, for a parameter space that allows for local non-exogeneity of the instruments but rules out weak instruments.