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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

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.

Suggested Citation

  • Jean-Marie Dufour, 2005. "Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and non-standard asymptotics," CIRANO Working Papers 2005s-02, CIRANO.
    • Handle: RePEc:cir:cirwor:2005s-02
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    References listed on IDEAS

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    1. Kiviet, Jan F. & Dufour, Jean-Marie, 1997. "Exact tests in single equation autoregressive distributed lag models," Journal of Econometrics, Elsevier, vol. 80(2), pages 325-353, October.
    2. Maxwell B. Stinchcombe & Halbert White, 1992. "Some Measurability Results for Extrema of Random Functions Over Random Sets," Review of Economic Studies, Oxford University Press, vol. 59(3), pages 495-514.
    3. Atsushi Inoue & Lutz Kilian, 2002. "Bootstrapping Autoregressive Processes with Possible Unit Roots," Econometrica, Econometric Society, vol. 70(1), pages 377-391, January.
    4. 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.
    5. DUFOUR, Jean-Marie & JOUINI, Tarek, 2005. "Finite-Sample Simulation-Based Inference in VAR Models with Applications to Order Selection and Causality Testing," Cahiers de recherche 2005-12, Universite de Montreal, Departement de sciences economiques.
    6. Campbell, Bryan & Dufour, Jean-Marie, 1997. "Exact Nonparametric Tests of Orthogonality and Random Walk in the Presence of a Drift Parameter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(1), pages 151-173, February.
    7. McFadden, Daniel, 1989. "A Method of Simulated Moments for Estimation of Discrete Response Models without Numerical Integration," Econometrica, Econometric Society, vol. 57(5), pages 995-1026, September.
    8. Gallant, A. Ronald & Tauchen, George, 1996. "Which Moments to Match?," Econometric Theory, Cambridge University Press, vol. 12(4), pages 657-681, October.
    9. 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(6), pages 944-961, December.
    10. 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 815-843, August.
    11. 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.
    12. Dufour, Jean-Marie, 1990. "Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors," Econometrica, Econometric Society, vol. 58(2), pages 475-494, March.
    13. 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.
    14. Jean-Marie Dufour & Abdeljelil Farhat & Lucien Gardiol & Lynda Khalaf, 1998. "Simulation-based finite sample normality tests in linear regressions," Econometrics Journal, Royal Economic Society, vol. 1(Conferenc), pages 154-173.
    15. 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.
    16. 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.
    17. Keane, Michael, 1993. "Simulation estimation for panel data models with limited dependent variables," MPRA Paper 53029, University Library of Munich, Germany.
    18. 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 335-355, March.
    19. repec:cup:etheor:v:12:y:1996:i:4:p:657-81 is not listed on IDEAS
    20. Joel L. Horowitz, 1996. "Bootstrap Methods in Econometrics: Theory and Numerical Performance," Econometrics 9602009, University Library of Munich, Germany, revised 05 Mar 1996.
    21. 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.
    22. 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.
    23. Lynda Khalaf & Jean-Marie Dufour, 2004. "Simulation-Based Finite-Sample Inference in Simultaneous Equations," Econometric Society 2004 North American Summer Meetings 239, Econometric Society.
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    More about this item

    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;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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