Under the null of valid specification, pre-tests of valid specification do not distort inference

Consider a parameter of interest, which can be consistently estimated under some conditions. Suppose also that we can at least partly test these conditions with specification tests. We consider the common practice of conducting inference on the parameter of interest conditional on not rejecting these tests. We show that if the tested conditions hold, conditional inference is valid but possibly conservative. This holds generally, without imposing any assumption on the asymptotic dependence between the estimator of the parameter of interest and the specification test.