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Autocovariance structure of powersof switching-regime ARMAProcesses

Published online by Cambridge University Press:  15 November 2002

Christian Francq  and
Jean-Michel Zakoïan
Christian Francq
Affiliation:
Université du Littoral-Côte d'Opale, LMPA J. Liouville, Centre Universitaire de la Mi-Voix, 50 rue F. Buisson, BP. 699, 62228 Calais Cedex, France; [email protected].
Jean-Michel Zakoïan
Affiliation:
Université de Lille 3 and CREST, 15 boulevard Gabriel Péri, 92245 Malakoff Cedex, France; [email protected].
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Abstract

In Francq and Zakoïan [4], we derived stationarity conditions forARMA(p,q) models subject to Markov switching. In this paper, weshow that, under appropriate moment conditions, the powers of thestationary solutions admit weak ARMA representations, which we areable to characterize in terms of p,q, the coefficients of themodel in each regime, and the transition probabilities of theMarkov chain. These representations are potentially useful forstatistical applications.

Keywords

ARMA representationhidden Markov modelsMarkov-switching modelsidentification.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 259 - 270
Copyright
© EDP Sciences, SMAI, 2002

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References

Baum, L.E. and Petrie, T., Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 30 (1966) 1554-1563. CrossRef
A. Berlinet, Estimation des degrés d'un ARMA multivarié, Pub. IRMA, Vol. 4. Lille (1982).
P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods. Springer-Verlag, New York (1991).
Francq, C. and Zakoïan, J.-M., Stationarity of Multivariate Markov-switching ARMA Models. J. Econometrics 102 (2001) 339-364. CrossRef
Hamilton, J.D., A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 (1989) 357-384. CrossRef
Hamilton, J.D., Specification testing in Markov switching time series models. J. Econometrics 45 (1996) 39-70. CrossRef
H. Karlsen, A class of non-linear time series models, Ph.D. Thesis. University of Bergen, Norway (1990).
Leroux, B.G. and Puterman, L.M., Maximum-penalized-likelihood estimation for independent and Markov-dependent mixture models. Biometrics 48 (1992) 545-558. CrossRef
Poskitt, D.S. and Chung, S.H., Markov chain models, time series analysis and extreme value theory. Adv. Appl. Probab. 28 (1996) 405-425. CrossRef
Robert, C.P., Rydén, T. and Titterington, D.M., Bayesian inference in hidden Markov models through the reversible jump Markov Chain Monte-Carlo method. J. Roy. Statist. Soc. B 62 (2000) 57-75. CrossRef
Rydén, T., Estimating the orders of hidden Markov models. Statistics 26 (1995) 345-354. CrossRef
Zhang, J. and Stine, R.A., Autocovariance structure of Markov regime switching models and model selection. J. Time Ser. Anal. 22 (2001) 107-124. CrossRef