Skip to main content Accessibility help
×
Home

ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION

Published online by Cambridge University Press:  07 October 2019

José Luis Montiel Olea
Affiliation:
Columbia University
Corresponding
E-mail address:

Abstract

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.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

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.

References

Abramowitz, M. & Stegun, I. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55, Dover Publications.Google Scholar
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, 4663.10.1214/aoms/1177730090CrossRefGoogle Scholar
Andrews, D., Moreira, M., & Stock, J. (2006) Optimal two-sided invariant similar tests for instrumental variables regression. Econometrica 74, 715752.10.1111/j.1468-0262.2006.00680.xCrossRefGoogle Scholar
Andrews, I. (2016) Conditional linear combination tests for weakly identified models. Econometrica 84, 21552182.10.3982/ECTA12407CrossRefGoogle Scholar
Chamberlain, G. (2007) Decision theory applied to an instrumental variables model. Econometrica 75, 609652.10.1111/j.1468-0262.2007.00764.xCrossRefGoogle Scholar
Chernozhukov, V., Hansen, C., & Jansson, M. (2009) Admissible invariant similar tests for instrumental variables regression. Econometric Theory 25, 806818.10.1017/S0266466608090312CrossRefGoogle Scholar
Ferguson, T. (1967) Mathematical Statistics: A Decision Theoretic Approach, vol. 7. Academic Press.Google Scholar
Le Cam, L. (1986) Asymptotic Methods in Statistical Decision Theory. Springer Verlag.10.1007/978-1-4612-4946-7CrossRefGoogle Scholar
Lehmann, E. & Romano, J. (2005) Testing Statistical Hypotheses. Springer Texts in Statistics. Springer Verlag.Google Scholar
Linnik, J. (1968). Statistical Problems with Nuisance Parameters, vol. 20. American Mathematical Society.Google Scholar
Moreira, M. (2003) A conditional likelihood ratio test for structural models. Econometrica 71, 10271048.10.1111/1468-0262.00438CrossRefGoogle Scholar
Moreira, M. & Moreira, H. (2015) Optimal Two-Sided Tests for Instrumental Variables Regression with Heteroskedastic and Autocorrelated Errors. Working paper, EPFGV.10.2139/ssrn.2608972CrossRefGoogle Scholar
Müller, U.K. (2011) Efficient tests under a weak convergence assumption. Econometrica 79, 395435.Google Scholar
Neyman, J. (1935) Sur la vérification des hypothèses statistiques composées. Bulletin de la Société Mathématique de France tome 63, 246266.10.24033/bsmf.1236CrossRefGoogle Scholar
Olver, F.W. (1997) Asymptotics and Special Functions, 2nd ed. AKP Classics, Academic press.10.1201/9781439864548CrossRefGoogle Scholar
Phillips, P.C. (1989) Partially identified econometric models. Econometric Theory 5, 181240.10.1017/S0266466600012408CrossRefGoogle Scholar
Staiger, D. & Stock, J. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557586.10.2307/2171753CrossRefGoogle Scholar
Stroock, D. (1999) A Concise Introduction to the Theory of Integration, 2nd ed. Birkhauser.Google Scholar
Wald, A. (1950) Statistical Decision Functions. Wiley.Google Scholar
Yogo, M. (2004) Estimating the elasticity of intertemporal substitution when instruments are weak. Review of Economics and Statistics 86, 797810.10.1162/0034653041811770CrossRefGoogle Scholar

Montiel Olea supplementary material

Montiel Olea supplementary material 1
File 943 KB

Montiel Olea et al. supplementary material

Montiel Olea et al. supplementary material 2
File 768 KB

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 109 *
View data table for this chart

* Views captured on Cambridge Core between 07th October 2019 - 21st January 2021. This data will be updated every 24 hours.

Hostname: page-component-76cb886bbf-frjnl Total loading time: 0.537 Render date: 2021-01-21T17:38:41.324Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *