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Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and non-standard asymptotics

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  • Jean-Marie Dufour

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Abstract

The technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] provides an attractive method of building exact tests from statistics whose finite sample distribution is intractable but can be simulated (provided it does not involve nuisance parameters). We extend this method in two ways: first, by allowing for MC tests based on exchangeable possibly discrete test statistics; second, by generalizing the method to statistics whose null distributions involve nuisance parameters (maximized MC tests, MMC). Simplified asymptotically justified versions of the MMC method are also proposed and it is shown that they provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics (e.g., unit root asymptotics). Parametric bootstrap tests may be interpreted as a simplified version of the MMC method (without the general validity properties of the latter). La technique des tests de Monte Carlo ((MC; Dwass (1957), Barnard (1963)) constitue une méthode attrayante qui permet de construire des tests exacts fondés sur des statistiques dont la distribution exacte est difficile à calculer par des méthodes analytiques mais peut être simulée, pourvu que cette distribution ne dépende pas de paramètres de nuisance. Nous généralisons cette méthode dans deux directions: premièrement, en considérant le cas où le test de Monte Carlo est construit à partir de réplications échangeables d’une variable aléatoire dont la distribution peut comporter des discontinuités; deuxièmement, en étendant la méthode à des statistiques dont la distribution dépend de paramètres de nuisance (tests de Monte Carlo maximisés, MMC). Nous proposons aussi des versions simplifiées de la procédure MMC, qui ne sont valides qu’asymptotiquement mais fournissent néanmoins une méthode simple qui permet d’améliorer les approximations asymptotiques usuelles, en particulier dans des cas non standards (e.g., l’asymptotique en présence de racines unitaires). Nous montrons aussi que les tests basés sur la technique du bootstrap paramétrique peut s’interpréter comme une version simplifiée de la procédure MMC. Cette dernière fournit toutefois des tests asymptotiquement valides sous des conditions beaucoup plus générales que le bootstrap paramétrique.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 2005s-02.

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Date of creation: 01 Feb 2005
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Handle: RePEc:cir:cirwor:2005s-02

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Keywords: Monte Carlo test; maximized monte Carlo test; finite sample test; exact test; nuisance parameter; bounds; bootstrap; parametric bootstrap; simulated annealing; asymptotics; nonstandard asymptotic distribution; test de Monte Carlo; test de Monte Carlo maximisé; test exact; test valide en échantillon fini; paramètre de nuisance; bornes; bootstrap; bootstrap paramétrique; recuit simulé; distribution asymptotique non standard;

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  1. Campbell, B. & Dufour, J.M., 1994. "Excat Nonparametric Tests of Orthogonality and Random Walk in the Presence of a Drift Parameter," Cahiers de recherche 9407, Universite de Montreal, Departement de sciences economiques.
  2. Alexander Benkwitz & Michael Neumann & Helmut Lutekpohl, 2000. "Problems related to confidence intervals for impulse responses of autoregressive processes," Econometric Reviews, Taylor & Francis Journals, vol. 19(1), pages 69-103.
  3. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
  4. Jean-Marie Dufour & Tarek Jouini, 2005. "Finite-Sample Simulation-Based Inference in VAR Models with Applications to Order Selection and Causality Testing," CIRANO Working Papers 2005s-26, CIRANO.
  5. Atsushi Inoue & Lutz Kilian, 2002. "Bootstrapping Autoregressive Processes with Possible Unit Roots," Econometrica, Econometric Society, vol. 70(1), pages 377-391, January.
  6. Dufour, Jean-Marie, 1989. "Nonlinear Hypotheses, Inequality Restrictions, and Non-nested Hypotheses: Exact Simultaneous Tests in Linear Regressions," Econometrica, Econometric Society, vol. 57(2), pages 335-55, March.
  7. Keane, Michael, 1993. "Simulation estimation for panel data models with limited dependent variables," MPRA Paper 53029, University Library of Munich, Germany.
  8. Goffe, William L. & Ferrier, Gary D. & Rogers, John, 1994. "Global optimization of statistical functions with simulated annealing," Journal of Econometrics, Elsevier, vol. 60(1-2), pages 65-99.
  9. Dufour, J.-M., 1986. "Exact tests and confidence sets in linear regressions with autocorrelated errors," CORE Discussion Papers 1986037, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  10. Joel L. Horowitz, 1996. "Bootstrap Methods in Econometrics: Theory and Numerical Performance," Econometrics 9602009, EconWPA, revised 05 Mar 1996.
  11. Daniel McFadden, 1987. "A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration," Working papers 464, Massachusetts Institute of Technology (MIT), Department of Economics.
  12. Dufour, J.M. & Kiviet, J.F., 1995. "Exact Tests in Single Equation Autoregressive Distributed Lag Models," Cahiers de recherche 9549, Universite de Montreal, Departement de sciences economiques.
  13. DUFOUR, Jean-Marie & FARHAT, Abdeljelil & GARDIOL, Lucien, 1998. "Simulation-Based Finite-Sample Normality Tests in Linear Regressions," Cahiers de recherche 9811, Universite de Montreal, Departement de sciences economiques.
  14. Gallant, A. Ronald & Tauchen, George, 1996. "Which Moments to Match?," Econometric Theory, Cambridge University Press, vol. 12(04), pages 657-681, October.
  15. Jouneau-Sion, Frederic & Torres, Olivier, 2006. "MMC techniques for limited dependent variables models: Implementation by the branch-and-bound algorithm," Journal of Econometrics, Elsevier, vol. 133(2), pages 479-512, August.
  16. Dufour, Jean-Marie & Jasiak, Joann, 2001. "Finite Sample Limited Information Inference Methods for Structural Equations and Models with Generated Regressors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 42(3), pages 815-43, August.
  17. Donald W. K. Andrews, 2000. "Inconsistency of the Bootstrap when a Parameter Is on the Boundary of the Parameter Space," Econometrica, Econometric Society, vol. 68(2), pages 399-406, March.
  18. Stinchcombe, Maxwell B & White, Halbert, 1992. "Some Measurability Results for Extrema of Random Functions over Random Sets," Review of Economic Studies, Wiley Blackwell, vol. 59(3), pages 495-514, July.
  19. repec:cup:etheor:v:12:y:1996:i:4:p:657-81 is not listed on IDEAS
  20. Marc Hallin & Jean-Marie Dufour & Ivan Mizera, 1998. "Generalized run tests for heteroscedastic time series," ULB Institutional Repository 2013/2077, ULB -- Universite Libre de Bruxelles.
  21. Inoue, Atsushi & Kilian, Lutz, 2003. "The Continuity Of The Limit Distribution In The Parameter Of Interest Is Not Essential For The Validity Of The Bootstrap," Econometric Theory, Cambridge University Press, vol. 19(06), pages 944-961, December.
  22. Lynda Khalaf & Jean-Marie Dufour, 2004. "Simulation-Based Finite-Sample Inference in Simultaneous Equations," Econometric Society 2004 North American Summer Meetings 239, Econometric Society.
  23. Dufour, Jean-Marie & Kiviet, Jan F., 1996. "Exact tests for structural change in first-order dynamic models," Journal of Econometrics, Elsevier, vol. 70(1), pages 39-68, January.
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