Conditional Inference with a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.