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ON MARKOV-SWITCHING ARMA PROCESSES—STATIONARITY, EXISTENCE OF MOMENTS, AND GEOMETRIC ERGODICITY

Published online by Cambridge University Press:  01 February 2009

Robert Stelzer
Robert Stelzer*
Affiliation:
Technische Universität München
*
*Address correspondence to Robert Stelzer, Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching, Germany; e-mail: rstelzer@ma.tum.de.
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Abstract

The probabilistic properties of ℝd-valued Markov-switching autoregressive moving average (ARMA) processes with a general state space parameter chain are analyzed. Stationarity and ergodicity conditions are given, and an easy-to-check general sufficient stationarity condition based on a tailor-made norm is introduced. Moreover, it is shown that causality of all individual regimes is neither a necessary nor a sufficient criterion for strict negativity of the associated Lyapunov exponent.

Finiteness of moments is also considered and geometric ergodicity and strong mixing are proven. The easily verifiable sufficient stationarity condition is extended to ensure these properties.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 1 , February 2009 , pp. 43 - 62
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

An, H.Z. & Huang, F.C. (1996) The geometrical ergodicity of nonlinear autoregressive models. Statistica Sinica 6, 943–956.Google Scholar
Ash, R.B. & Gardner, M.F. (1975) Topics in Stochastic Processes. Academic Press.Google Scholar
Bauer, H. (1992) Maß- und Integrationstheorie, 2nd ed.de Gruyter.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714–1730.Google Scholar
Brandt, A. (1986) The stochastic equation Yn +1 = AnYn + Bn with stationary coefficients. Advances in Applied Probability 18, 211–220.Google Scholar
Brandt, A., Franken, P. & Lisek, B. (1990) Stationary Stochastic Models. Wiley.Google Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed.Springer-Verlag.CrossRefGoogle Scholar
Douc, R., Moulines, E., & Rydén, T. (2004) Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Annals of Statistics 32, 2254–2304.CrossRefGoogle Scholar
Doucet, A., Logothetis, A. & Krishnamurthy, V. (2000) Stochastic sampling algorithms for state estimation of jump Markov linear systems. IEEE Transactions on Automatic Control 45, 188–202.CrossRefGoogle Scholar
Feigin, P.D. & Tweedie, R.L. (1985) Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. Journal of Time Series Analysis 6, 1–14.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2001) Stationarity of multivariate Markov-switching ARMA models. Journal of Econometrics 102, 339–364.CrossRefGoogle Scholar
Furstenberg, H. & Kesten, H. (1960) Products of random matrices. Annals of Mathematical Statistics 31, 457–469.Google Scholar
Goldie, C.M. & Maller, R.A. (2000) Stability of perpetuities. Annals of Probability 28, 1195–1218.CrossRefGoogle Scholar
Hamilton, J.D. (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.Google Scholar
Hamilton, J.D. (1990) Analysis of time series subject to changes in regime. Journal of Econometrics 45, 39–70.Google Scholar
Hamilton, J.D. & Raj, B. (eds.) (2002) Advances in Markov-Switching Models—Applications in Business Cycle Research and Finance. Physica-Verlag.Google Scholar
Hille, E. & Phillips, R.S. (1957) Functional Analysis and Semi-Groups, rev. ed. American Mathematical Society.Google Scholar
Karlsen, H.A. (1990) Existence of moments in a stationary stochastic difference equation. Advances in Applied Probability 22, 129–146.CrossRefGoogle Scholar
Klüppelberg, C. & Pergamenchtchikov, S. (2004) The tail of the stationary distribution of a random coefficient AR(q) model. Annals of Applied Probability 14, 971–1005.Google Scholar
Krengel, U. (1985) Ergodic Theorems. de Gruyter.CrossRefGoogle Scholar
Krolzig, H.-M. (1997) Markov-Switching Vector Autoregressions. Lecture Notes in Economics and Mathematical Systems 454. Springer-Verlag.CrossRefGoogle Scholar
Lee, O. (2005) Probabilistic properties of a nonlinear ARMA process with Markov switching. Communications in Statistics—Theory and Methods 34, 193–204.Google Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.CrossRefGoogle Scholar
Nicholls, D.F. & Quinn, B.G. (1982) Random Coefficient Autoregressive Models: An Introduction. Lecture Notes in Statistics 11. Springer-Verlag.CrossRefGoogle Scholar
Resnick, S.I. (1992) Adventures in Stochastic Processes. Birkhäuser.Google Scholar
Stelzer, R. (2008) Multivariate Markov-switching ARMA processes with regularly varying noise. Journal of Multivariate Analysis 99, 1177–1190.CrossRefGoogle Scholar
Tugnait, J.K. (1982) Adaptive estimation and identification for discrete systems with Markov jump parameters. IEEE Transactions on Automatic Control 27, 1054–1065. Correction Published in 1984, vol. 29, 1984, p. 286.Google Scholar
Yao, J. (2001) On square-integrability of an AR process with Markov switching. Statistics and Probability Letters 52, 265–270.CrossRefGoogle Scholar
Yao, J.F. & Attali, J.G. (2000) On stability of nonlinear AR processes with Markov switching. Advances in Applied Probability 32, 394–407.Google Scholar