Is Inference Conditional on Not Rejecting a Pre-test Less Reliable than Unconditional Inference?

Assume that an estimator is asymptotically normal for a target parameter under some conditions. Suppose also that one can test these conditions, and one conducts inference for the target only if the pre-test is not rejected. Does such pre-testing undermine inference? We show that if the tested conditions and mild regularity restrictions hold, conditional inference is still valid, albeit typically conservative. Validity holds regardless of the asymptotic dependence between the estimator and the pre-test. If the tested conditions do not hold, we exhibit conditions under which confidence intervals have larger conditional than unconditional coverage.