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On stability of nonlinear AR processes with Markov switching

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

J.-F. Yao  and
J.-G. Attali
J.-F. Yao*
Affiliation:
SAMOS, Université Paris 1
J.-G. Attali*
Affiliation:
Université Paris 6 and SAMOS, Université Paris 1
*
∗ Postal address: SAMOS, Université Paris 1, 90, rue de Tolbiac, F-75634 Paris cedex 13, France. Email address: yao@univ-paris1.fr
∗∗ Postal address: Labo. de Probabilités, Université Paris 6, 4, place Jussieu, F-75252 Paris cedex 05, France.
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Abstract

We investigate the stability problem for a nonlinear autoregressive model with Markov switching. First we give conditions for the existence and the uniqueness of a stationary ergodic solution. The existence of moments of such a solution is then examined and we establish a strong law of large numbers for a wide class of unbounded functions, as well as a central limit theorem under an irreducibility condition.

Keywords

Markov switchingnonlinear ARLyapounov functionsstability

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 394 - 407
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Attali, J.-G. (1999). (1). Méthodes de stabilité pour des chaines de Markov non felleriennes. (2). Sur quelques autres problèmes issus des réseaux de neurones. Thèse, Université Paris I, France.Google Scholar
[2] Bhattacharya, R. N. and Lee, C. (1995). Ergodicity of nonlinear first order autoregressive models. J. Theor. Prob. 8, 207–219.CrossRefGoogle Scholar
[3] Borovkov, A. (1992). On ergodicity of iterations of random {Lipschitz} transformations. Russian Acad. Sci. Dokl. Math. 45, 551–555.Google Scholar
[4] Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM J. Control Optimiz. 31, 942–959.CrossRefGoogle Scholar
[5] Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Prob. 20, 1714–1730.CrossRefGoogle Scholar
[6] Brandt, A. (1986). The stochastic equation Y_{n+1}=A_{n} Y_{n}+ B_{n} with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
[7] Duflo, M. (1997). Random Iterative Models. Springer, New York.Google Scholar
[8] Francq, C. and Roussignol, M. (1998). Ergodicity of autoregressive processes with Markov switching and consistency of the maximum-likelihood estimator. Consistency of MLE for AR models with Markov switching. Prépublication No. 10/96, Université de Marne-La-Vallée. Statistics 32, 151–173.CrossRefGoogle Scholar
[9] Gallant, A. R. (1987). Nonlinear Statistical Models. John Wiley, New York.CrossRefGoogle Scholar
[10] Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.CrossRefGoogle Scholar
[11] Hamilton, J. D. (1990). Analysis of time series subject to changes in regime. J. Economet. 45, 39–70.CrossRefGoogle Scholar
[12] Hamilton, J. D. (1993). Estimation, inference and forecasting of time series subject to changes in regime. In Handbook of Statistics, Vol. 11, eds Maddala, G., Rao, C. R. and Vinod, H. D. Elsevier, Amsterdam, pp. 230–260.Google Scholar
[13] Hamilton, J. D. (1996). Specification testing in Markov-switching time-series models. J. Economet. 70, 127–157.CrossRefGoogle Scholar
[14] Holst, U., Lingren, G., Holst, J. and Thuvesholmen, M. (1994). Recursive estimation in switching autoregressions with a Markov regime. J. Time Ser. Anal. 15, 489–506.CrossRefGoogle Scholar
[15] Kugler, P. (1996). The term structure of interest rates and regime shifts: some empirical results. Economics Lett. 50, 121–126.CrossRefGoogle Scholar
[16] Lindvall, T. (1992). Lectures on Coupling Methods. John Wiley, New York.Google Scholar
[17] MacDonald, I. L. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-valued Time Series. Chapman and Hall, New York.Google Scholar
[18] McCulloch, R. E. and Tsay, R. S. (1994). Statistical analysis of econometric time series via Markov switching models. J. Time Ser. Anal. 15, 523–539.CrossRefGoogle Scholar
[19] Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.CrossRefGoogle Scholar
[20] Perraudin, C. (1999). Asymétries conjonctuelles and dynamique de l'emploi: essai de modélisations non-linéaires. Thèse, Université Paris I, France.Google Scholar
[21] Pötscher, B. M. and Prucha, L. R. (1997). Dynamic Nonlinear Econometric Models. Springer, Berlin.CrossRefGoogle Scholar
[22] Tanikawa, A. (1999). Geometric ergodicity of nonlinear first order autoregressive models. Commun. Stat. Stoch. Models 15, 227–245.CrossRefGoogle Scholar
[23] Tjöstheim, D., (1990). Nonlinear time series and Markov chains. Adv. Appl. Prob. 22, 587–611.CrossRefGoogle Scholar
[24] Tong, H. (1990). Non-linear Time Series (A Dynamic System Approach). Oxford University Press, New York.CrossRefGoogle Scholar
[25] Yao, J. (1998). On least squares estimation for stable nonlinear autoregressive processes. To appear in Ann. Inst. Math. Statist.Google Scholar