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On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Shiqing Ling
Shiqing Ling*
Affiliation:
University of Western Australia
*
∗Postal address: Department of Economics, University of Western Australia, Nedlands, Western Australia 6907, Australia. Email address: [email protected]
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Abstract

Following Tweedie (1988), this paper constructs a special test function which leads to sufficient conditions for the stationarity and finiteness of the moments of a general non-linear time series model, the double threshold ARMA conditional heteroskedastic (DTARMACH) model. The results are applied to two well-known special cases, the GARCH and threshold ARMA (TARMA) models. The condition for the finiteness of the moments of the GARCH model is simple and easier to check than the condition given by Milhøj (1985) for the ARCH model. The condition for the stationarity of the TARMA model is identical to the condition given by Brockwell et al. (1992) for a special case, and verifies their conjecture that the moving average component does not affect the stationarity of the model. Under an additional irreducibility assumption, the geometric ergodicity of the GARCH and TARMA models is also established.

Keywords

DTARMACH modelfiniteness of momentsGARCH modelMarkov chainstrict stationarityTARMA model

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 688 - 705
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This paper is drawn from the second chapter of my Ph.D. dissertation under the supervision of Professor W. K. Li at The University of Hong Kong.

References

An, H. Z., and Huang, F. C. (1996). The geometric ergodicity of nonlinear autoregressive models. Statist. Sinica 6, 943–956.Google Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T., Chou, R. Y. and Kroner, K. F. (1992). ARCH modelling in finance. J. Econometrics 52, 5–59.CrossRefGoogle Scholar
Brockwell, P. J., Liu, J., and Tweedie, R. L. (1992). On the existence of stationary threshold autoregressive moving-average processes. J. Time Ser. Anal. 2, 95–107.CrossRefGoogle Scholar
Chan, K. S., and Tong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. Appl. Prob. 17, 666–678.CrossRefGoogle 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
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Feigin, P. D, and Tweedie, R. L. (1985). Random coefficient autoregressive processes. J. Time Ser. Anal. 6, 1–14.CrossRefGoogle Scholar
Guegan, D., and Diebolt, J. (1994). Probabilistic properties of the β-ARCH model. Statist. Sinica 4, 71–89.Google Scholar
Li, C. W., and Li, W. K. (1996). On a double threshold autoregressive conditional heteroskedastic time series model. J. Appl. Econometrics 11, 253–274.3.0.CO;2-8>CrossRefGoogle Scholar
Ling, S., and Li, W. K. (1997). Fractional ARIMA-GARCH time series models. J. Amer. Statist. Assoc. 92, 1184–1194.CrossRefGoogle Scholar
Liu, J. (1989). A simple condition for the existence of some stationary bilinear time series. J. Time Ser. Anal. 1, 33–39.CrossRefGoogle Scholar
Liu, J. and Susko, Ed. (1992). On strict stationarity and ergodicity of a non-linear ARMA model. J. Appl. Prob. 29, 363–373.CrossRefGoogle Scholar
Liu, J., Li, W. K., and Li, C. W. (1997). On a threshold autoregression with conditional heteroskedastic variances. J. Statist. Plann. Inference, 2, 279–300.CrossRefGoogle Scholar
Milhöj, A. (1985). The moment structure of ARCH processes. Scand. J. Statist. 12, 281–292.Google Scholar
Nummelin, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Pantula, S. G. (1989). Estimation of autoregressive models with ARCH errors. Sankhya B 50, 119–138.Google Scholar
Tjöstheim, D. (1990). Non-linear time series and Markov chains. Adv. Appl. Prob. 22, 587–611.CrossRefGoogle Scholar
Tong, H. (1978). On a threshold model. In Pattern Recognition and Signal Processing. Ed. Chen, C. H. Sijthoff and Noordhoff, Holland.Google Scholar
Tong, H. (1990). Nonlinear Time Series: a Dynamical System Approach. Oxford University Press, Oxford.CrossRefGoogle 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.CrossRefGoogle Scholar
Tweedie, R. L. (1983). Criteria for rates of convergence of Markov chains, with application to queueing theory. In Papers in Probability, Statistics and Analysis. Ed. Kingman, J. F. C. and Reuter, G. E. H. Cambridge University Press, Cambridge.Google Scholar
Tweedie, R. L. (1988). Invariant measure for Markov chains with no irreducibility assumptions. J. Appl. Prob. 25A, 275–285.CrossRefGoogle Scholar
Weiss, A. A. (1986). Asymptotic theory on ARCH models: estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar