Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T19:34:17.217Z Has data issue: false hasContentIssue false
  • English
  • Français

On strict stationarity and ergodicity of a non-linear ARMA model

Part of: Markov processes Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jian Liu  and
Ed Susko
Jian Liu*
Affiliation:
University of British Columbia
Ed Susko*
Affiliation:
University of British Columbia
*
∗Postal address for both authors: Department of Statistics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada.
∗Postal address for both authors: Department of Statistics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

Two recent papers by Petruccelli and Woolford (1984) and Chan et al. (1985) showed that the key element governing ergodicity of a threshold AR(1) model is the joint behavior of the two linear AR(1) pieces falling in the two boundary threshold regimes. They used essentially the necessary and sufficient conditions for ergodicity of a general Markov chain of Tweedie (1974), (1975) in a rather clever manner. However, it is difficult to extend the results to the more general threshold ARMA models. Besides, irreducibility is also required to apply Tweedie's results. In this paper, instead of pursuing the ideas in Tweedie's results, we shall develop a criterion similar in spirit to the technique used by Beneš (1967) in the context of continuous-time Markov chains. Consequently, we derive a necessary and sufficient condition for existence of a strictly stationary solution of a general non-linear ARMA model to be introduced in Section 2 of this paper. This condition is then applied to the threshold ARMA(1, q) model to yield a sufficient condition for strict stationarity which is identical to the condition given by Petruccelli and Woolford (1984) for the threshold AR(1). Hence, the conjecture that the moving average component does not affect stationarity is partially verified. Furthermore, under an additional irreducibility assumption, ergodicity of a non-linear ARMA model is established. The paper then concludes with a necessary condition for stationarity of the threshold ARMA(1, q) model.

Keywords

DISCRETE-TIME MARKOV CHAINS

MSC classification

Primary: 62M05: Markov processes
Secondary: 60G10: Stationary processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 363 - 373
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beneš, V. E. (1967) Existence of finite invariant measures for Markov processes. Proc. Amer. Math. Soc. 18, 1058–1061.CrossRefGoogle Scholar
Brockwell, P. J., Liu, J. and Tweedie, R. L. (1990) On the existence of stationary threshold ARMA processes, to appear in J. Time Series Anal. Google Scholar
Chan, K. S., Petruccelli, J. D., Tong, H. and Woolford, S. W. (1985) A multiple-threshold AR(1) model. J. Appl. Prob. 22, 267–279.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T. (1958) Linear Operators, Part I. Interscience, New York.Google Scholar
Petruccelli, J. D. and Woolford, S. W. (1984) A threshold AR(1) model. J. Appl. Prob. 21, 270–286.CrossRefGoogle Scholar
Tong, H. (1983) Threshold Models in Non-Linear Time Series Analysis. Lecture Notes in Statistics 21, Springer-Verlag, New York.Google Scholar
Tweedie, R. L. (1974) R-theory for Markov chains on a general state space I: solidarity properties and R-recurrent chains. Ann. Prob. 2, 840–864.Google Scholar
Tweedie, R. L. (1975) Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stoch. Proc. Appl. 3, 385–403.Google Scholar