Conditional Inference with a Functional Nuisance Parameter

TL;DR

This paper develops a new approach for hypothesis testing in moment condition models with an infinite-dimensional nuisance parameter, using conditional tests that are asymptotically valid across various models.

Contribution

It introduces a sufficient statistic for the nuisance parameter and proposes conditional tests that are uniformly correct in size, enhancing testing robustness.

Findings

Conditional tests have uniformly correct asymptotic size.

Quasi-likelihood ratio based tests are efficient in strongly identified models.

The approach performs well compared to existing methods in examples.

Abstract

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.