Published online by Cambridge University Press: 07 October 2019
This article studies a classical problem in statistical decision theory: a hypothesis test of a sharp null in the presence of a nuisance parameter. The main contribution of this article is a characterization of two finite-sample properties often deemed reasonable in this environment: admissibility and similarity. Admissibility means that a test cannot be improved uniformly over the parameter space. Similarity requires the null rejection probability to be unaffected by the nuisance parameter.
The characterization result has two parts. The first part—established by Chernozhukov, Hansen, and Jansson (2009, Econometric Theory 25, 806–818)—states that maximizing weighted average power (WAP) subject to a similarity constraint suffices to generate admissible, similar tests. The second part—hereby established—states that constrained WAP maximization is (essentially) a necessary condition for a test to be admissible and similar. The characterization result shows that choosing an admissible, similar test is tantamount to selecting a particular weight function to report weighted average power. This result applies to full vector inference with a nuisance parameter, not to subvector inference.
The article also revisits the theory of testing in the instrumental variables model. Specifically—and in light of the relevance of the weighted average power criterion in the main theoretical result—the article suggests a weight function for the structural parameters of the homoskedastic instrumental variables model, based on the priors proposed by Chamberlain (2007). The corresponding test is, by construction, admissible and similar. In addition, the test is shown to have finite- and large-sample properties comparable to those of the conditional likelihood ratio test.
A previous version of this article was circulated under the title “Efficient Conditionally Similar-on-the-Boundary Tests”. I am deeply indebted to my main advisers Matías Cattaneo, Gary Chamberlain, Tomasz Strzalecki, and James Stock, for their continuous guidance, support, patience, and encouragement. I would like to thank seminar audiences at Banco de México, Brown, Chicago Booth (Econometrics), Cowles Summer Conference, Davis ARE, Duke, Imperial College Business School, ITAM, Northwestern, University of Chicago, University of Michigan, NYU Economics, Penn State, and TSE. I owe special thanks to Isaiah Andrews, Michael Jansson, Frank Schorfheide, Anna Mikusheva, Peter Phillips, Elie Tamer, Quang Vuong, and four anonymous referees for extremely helpful comments and suggestions. I would also like to thank Luigi Caloi, Hamza Husain, and Amilcar Velez for excellent research assistance. All errors are my own. First draft: September 18th, 2013. This version: July 8th, 2019.