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Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models

Part of: Inference from stochastic processes Ergodic theory Mathematical economics Markov processes

Published online by Cambridge University Press:  01 July 2016

Daren B. H. Cline
Daren B. H. Cline*
Affiliation:
Texas A&M University
*
∗ Postal address: Department of Statistics, Texas A&M University, College Station TX 77843-3143, USA. Email address: dcline@stat.tamu.edu
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Abstract

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Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

Keywords

Feller operatorT-chainGARCHthreshold time series

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 37A50: Relations with probability theory and stochastic processes 62M10: Time series, auto-correlation, regression, etc. 91B84: Economic time series analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 49 - 76
Copyright
Copyright © Applied Probability Trust 2011 

References

Ash, R. B. (1972). Real Analysis and Probability. Academic Press, New York.Google Scholar
Cline, D. B. H. (2007). Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models. Adv. Appl. Prob. 39, 462–491. (Correction: 39 (2007), 1115–1116.)Google Scholar
Li, C. W. and Li, W. K. (1996). On a double-threshold autoregressive heteroscedastic time series model. J. Appl. Econometrics 11, 253–274.Google Scholar
Ling, S. (1999). On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. J. Appl. Prob. 36, 688–705.CrossRefGoogle Scholar
Ling, S. (2007). A double AR(p) model: structure and estimation. Statistica Sinica 17, 161–175.Google Scholar
Ling, S. and McAleer, M. (2002). Stationarity and the existence of a family of GARCH processes. J. Econometrics 106, 109–117.Google Scholar
Liu, J., Li, W. K. and Li, C. W. (1997). On a threshold autoregression with conditional heteroscedastic variances. J. Statist. Planning Infer. 62, 279–300.Google Scholar
Lu, Z. (1996). A note on the geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model. Statist. Prob. Lett. 30, 305–311.Google Scholar
Meitz, M. and Saikkonen, P. (2008). Stability of nonlinear AR-GARCH models. J. Time Ser. Anal. 29, 453–475.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Munkres, J. R. (1975). Topology: A First Course. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Nummelin, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press.Google Scholar