Locally Regular and Efficient Tests in Non-Regular Semiparametric Models

TL;DR

This paper develops a framework for hypothesis testing in non-regular semiparametric models, establishing local regularity of C(α) tests and characterizing their optimality even when regular estimators are unavailable.

Contribution

It introduces a new approach to hypothesis testing in non-regular semiparametric models, including models with weak identification, and extends classical power bounds to this setting.

Findings

C(α) tests are locally regular under mild conditions

Power bounds are generalized for non-regular models

Application to single index and instrumental variables models

Abstract

This paper considers hypothesis testing in semiparametric models which may be non-regular. I show that C($\alpha$) style tests are locally regular under mild conditions, including in cases where locally regular estimators do not exist, such as models which are (semiparametrically) weakly identified. I characterise the appropriate limit experiment in which to study local (asymptotic) optimality of tests in the non-regular case and generalise classical power bounds to this case. I give conditions under which these power bounds are attained by the proposed C($\alpha$) style tests. The application of the theory to a single index model and an instrumental variables model is worked out in detail.