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Asymptotic Null Distribution of the Likelihood Ratio Test in Markov Switching Models

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  • René Garcia

Abstract

The Markov Switching Model, introduced by Hamilton (1988, 1989), has been used in various economic and financial applications where changes in regime play potentially an important role. While estimation methods for these models are by now well established, such is not the case for the corresponding testing procedures. The Markov switching models raise a special problem known in the statistics literature as testing hypotheses in models where a nuisance parameter is not identified under the null hypothesis. In these circumstances, the asymptotic distributions of the usual tests (likelihood ratio, Lagrange multiplier, Wald tests) are non-standard. In this paper, we show that, if we treat the transition probabilities as nuisance parameters in a Markov switching model and set the null hypothesis in terms uniquely of the parameters governed by the Markov variable, the distributional theory proposed by Hansen (1991a) is applicable to Markov switching models under certain assumptions. Based on this framework, we derive analytically, in the context of two-state Markov switching models, the asymptotic null distribution of the likelihood ratio test (which is shown to be also valid for the Lagrange multiplier and Wald tests under certain conditions) and the related covariance functions. Monte Carlo simulations show that the asymptotic distributions offer a very good approximation to the corresponding empirical distributions. Les modèles à changements de régime markoviens soulèvent un problème particulier connu dans la littérature statistique sous la rubrique des tests d'hypothèse dans les modèles où un paramètre de nuisance n'est pas identifié sous l'hypothèse nulle. Dans ces cas, les distributions asymptotiques des tests usuels (ratio de vraisemblance, multiplicateur de Lagrange, Wald) ne sont pas standard. Dans le présent article, nous montrons que, si nous traitons les probabilités de transition comme des paramètres de nuisance dans un modèle à changements de régime markoviens et fixons l'hypothèse nulle uniquement en fonction des paramètres régis par la variable de Markov, la théorie distributionnelle proposée par Hansen (1991) est applicable aux modèles à changements de régime markoviens sous certaines hypothèses. Dans ce cadre, nous dérivons analytiquement la distribution asymptotique du ratio de vraisemblance sous l'hypothèse nulle ainsi que les fonctions de covariance correspondantes pour divers modèles à changements de régime markoviens : un modèle à 2 moyennes avec erreurs non corrélées et homoscédastiques; un modèle à 2 moyennes avec des erreurs suivant un processus AR(r) ; et finalement un modèle à 2 moyennes et 2 variances avec des erreurs non corrélées. Dans les trois cas, des expériences de Monte Carlo montrent que les distributions asymptotiques dérivées offrent une très bonne approximation de la distribution empirique. La dérivation de la distribution asymptotique de la statistique du ratio de vraisemblance pour ces trois modèles simples markoviens à 2 états sera utile pour évaluer la signification statistique des résultats qui sont apparus dans la littérature et plus généralement pour offrir un ensemble de valeurs critiques aux chercheurs futurs dans ce domaine. Les valeurs critiques de la distribution asymptotique du test du ratio de vraisemblance dans les modèles à changements de régime markoviens sont considérablement plus élevées que les valeurs critiques impliquées par la distribution chi-carré standard.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 95s-07.

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Date of creation: 01 Feb 1995
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Handle: RePEc:cir:cirwor:95s-07

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Keywords: Markov switching models ; Transition probabilities; Null hypothesis; Modèles à changements de régime markoviens ; Probabilités de transition ; Hypothèse nulle;

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  1. Cecchetti, Stephen G & Lam, Pok-sang & Mark, Nelson C, 1990. "Mean Reversion in Equilibrium Asset Prices," American Economic Review, American Economic Association, vol. 80(3), pages 398-418, June.
  2. Robert B. Davies, 2002. "Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case," Biometrika, Biometrika Trust, vol. 89(2), pages 484-489, June.
  3. Garcia, Rene & Perron, Pierre, 1996. "An Analysis of the Real Interest Rate under Regime Shifts," The Review of Economics and Statistics, MIT Press, vol. 78(1), pages 111-25, February.
  4. Turner, Christopher M. & Startz, Richard & Nelson, Charles R., 1989. "A Markov model of heteroskedasticity, risk, and learning in the stock market," Journal of Financial Economics, Elsevier, vol. 25(1), pages 3-22, November.
  5. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-30, March.
  6. Donald W.K. Andrews, 1990. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Cowles Foundation Discussion Papers 943, Cowles Foundation for Research in Economics, Yale University.
  7. Hamilton, James D., 1988. "Rational-expectations econometric analysis of changes in regime : An investigation of the term structure of interest rates," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 385-423.
  8. Russell Davidson & James G. MacKinnon, 1980. "Several Tests for Model Specification in the Presence of Alternative Hypotheses," Working Papers 378, Queen's University, Department of Economics.
  9. Engel, Charles & Hamilton, James D, 1990. "Long Swings in the Dollar: Are They in the Data and Do Markets Know It?," American Economic Review, American Economic Association, vol. 80(4), pages 689-713, September.
  10. Donald W.K. Andrews & Werner Ploberger, 1993. "Admissibility of the Likelihood Ratio Test When a Nuisance Parameter Is Present OnlyUnder the Alternative," Cowles Foundation Discussion Papers 1058, Cowles Foundation for Research in Economics, Yale University.
  11. Turner, C.M. & Startz, R. & Nelson, C.R., 1989. "The Markov Model Of Heteroskedasticity, Risk And Learning In The Stock Market," Working Papers 89-01, University of Washington, Department of Economics.
  12. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-84, March.
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