Advances in Applied Probability

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

Daren B. H. Cline

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 43, Number 1 (2011), 49-76.

Dates
First available in Project Euclid: 15 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aap/1300198512

Digital Object Identifier
doi:10.1239/aap/1300198512

Mathematical Reviews number (MathSciNet)
MR2761147

Zentralblatt MATH identifier
1238.62101

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 91B84: Economic time series analysis [See also 62M10]

Keywords
Feller operator T-chain GARCH threshold time series

Citation

Cline, Daren B. H. Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models. Adv. in Appl. Probab. 43 (2011), no. 1, 49--76. doi:10.1239/aap/1300198512. https://projecteuclid.org/euclid.aap/1300198512


Export citation

References

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