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

Part of: Inference from stochastic processes 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.

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