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  • José Luis Montiel Olea (a1)


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.


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*Address correspondence to José Luis Montiel Olea, Department of Economics, Columbia University, 1022 International Affairs Building (IAB), 420 West 118th Street, New York, NY 10027, USA; e-mail:


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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.



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Abramowitz, M. & Stegun, I. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55, Dover Publications.
Anderson, T. & Rubin, H. (1949) Estimation of the parameters of a single equation in a complete system of stochastic equations. The Annals of Mathematical Statistics 20, 4663.
Andrews, D., Moreira, M., & Stock, J. (2006) Optimal two-sided invariant similar tests for instrumental variables regression. Econometrica 74, 715752.
Andrews, I. (2016) Conditional linear combination tests for weakly identified models. Econometrica 84, 21552182.
Chamberlain, G. (2007) Decision theory applied to an instrumental variables model. Econometrica 75, 609652.
Chernozhukov, V., Hansen, C., & Jansson, M. (2009) Admissible invariant similar tests for instrumental variables regression. Econometric Theory 25, 806818.
Ferguson, T. (1967) Mathematical Statistics: A Decision Theoretic Approach, vol. 7. Academic Press.
Le Cam, L. (1986) Asymptotic Methods in Statistical Decision Theory. Springer Verlag.
Lehmann, E. & Romano, J. (2005) Testing Statistical Hypotheses. Springer Texts in Statistics. Springer Verlag.
Linnik, J. (1968). Statistical Problems with Nuisance Parameters, vol. 20. American Mathematical Society.
Moreira, M. (2003) A conditional likelihood ratio test for structural models. Econometrica 71, 10271048.
Moreira, M. & Moreira, H. (2015) Optimal Two-Sided Tests for Instrumental Variables Regression with Heteroskedastic and Autocorrelated Errors. Working paper, EPFGV.
Müller, U.K. (2011) Efficient tests under a weak convergence assumption. Econometrica 79, 395435.
Neyman, J. (1935) Sur la vérification des hypothèses statistiques composées. Bulletin de la Société Mathématique de France tome 63, 246266.
Olver, F.W. (1997) Asymptotics and Special Functions, 2nd ed. AKP Classics, Academic press.
Phillips, P.C. (1989) Partially identified econometric models. Econometric Theory 5, 181240.
Staiger, D. & Stock, J. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557586.
Stroock, D. (1999) A Concise Introduction to the Theory of Integration, 2nd ed. Birkhauser.
Wald, A. (1950) Statistical Decision Functions. Wiley.
Yogo, M. (2004) Estimating the elasticity of intertemporal substitution when instruments are weak. Review of Economics and Statistics 86, 797810.
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  • José Luis Montiel Olea (a1)


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