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Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models

Part of: Stochastic processes Inference from stochastic processes 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: [email protected]
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Abstract

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We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Keywords

Stochastic recursionGARCH modelergodicitystationary distribution

MSC classification

Primary: 60G10: Stationary processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62M10: Time series, auto-correlation, regression, etc. 91B84: Economic time series analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 462 - 491
Copyright
Copyright © Applied Probability Trust 2007 

References

Alsmeyer, G. and Fuh, C.-D. (2001). Limit theorems for iterated random functions by regenerative methods. Stoch. Process. Appl. 96, 123–142.CrossRefGoogle Scholar
Barnsley, M. F. and Elton, J. H. (1988). A new class of Markov processes for image encoding. Adv. Appl. Prob. 20, 14–32.CrossRefGoogle Scholar
Basrak, B., Davis, R. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.CrossRefGoogle Scholar
Borkovec, M. and Klüppelberg, C. (2001). The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann. Appl. Prob. 11, 1220–1241.CrossRefGoogle Scholar
Bougerol, P. and Picard, N. (1992a). Stationarity of GARCH processes and some nonnegative time series. J. Econometrics 52, 115–127.Google Scholar
Bougerol, P. and Picard, N. (1992b). Strict stationarity of generalized autoregressive processes. Ann. Prob. 20, 1714–1730.Google Scholar
Brandt, A. (1986). The stochastic equation Y n+1=A n Y n B n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google 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
Cline, D. B. H. (2007a). Evaluating the Lyapunov exponent and existence of moments for threshold AR-ARCH models. J. Time Series Anal. 28, 241–260.Google Scholar
Cline, D. B. H. (2007b). Regular variation of order 1 nonlinear AR-ARCH models. Stoch. Process. Appl. 117, 840–861.CrossRefGoogle Scholar
Cline, D. B. H. and Pu, H. H. (1999). Geometric ergodicity of nonlinear time series. Statist. Sinica 9, 1103–1118.Google Scholar
Cline, D. B. H. and Pu, H. H. (2001). Geometric transience of nonlinear time series. Statist. Sinica 11, 273–287.Google Scholar
Cline, D. B. H. and Pu, H. H. (2004). Stability and the Lyaponouv exponent of threshold AR-ARCH models. Ann. Appl. Prob. 14, 1920–1920.Google Scholar
De Saporta, B. (2005). Tail of the stationary solution of the stochastic equation Y n+1=a n Y n b n with Markovian coefficients. Stoch. Process. Appl. 115, 1954–1954.CrossRefGoogle Scholar
Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45–76.CrossRefGoogle Scholar
Elton, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stoch. Process. Appl. 34, 39–47.Google Scholar
Franq, C. and Zakoïan, J.-M. (2005). The L 2-structures of standard and switching-regime GARCH models. Stoch. Process. Appl. 115, 1577–1582.Google Scholar
Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann Prob. 28, 1195–1218.CrossRefGoogle Scholar
Grincevičius, A. K. (1981). A random difference equation. Lithuanian Math. J. 20, 279–282.Google Scholar
He, C. and Teräsvirta, T. (1999a). Fourth moment structure of the GARCH(p,q) process. Econom. Theory 15, 824–846.CrossRefGoogle Scholar
He, C. and Teräsvirta, T. (1999b). Properties of moments of a family of GARCH processes. J. Econometrics 92, 173–192.CrossRefGoogle Scholar
Hwang, S. Y. and Woo, M.-J. (2001). Threshold ARCH(1) processes: asymptotic inference. Statist. Prob. Lett. 53, 11–20.CrossRefGoogle Scholar
Jarner, S. F. and Tweedie, R. L. (2001). Locally contractive iterated functions and stability of Markov chains. J. Appl. Prob. 38, 494–507.CrossRefGoogle Scholar
Klüppelberg, C. and Pergamenchtchikov, P. (2004). The tail of the stationary distribution of a random coefficient AR(q) model. Ann. Appl. Prob. 14, 971–1005.CrossRefGoogle Scholar
Lanne, M. and Saikkonen, P. (2005). Nonlinear GARCH models for highly persistent volatility. Econom. J. 8, 251–276.CrossRefGoogle Scholar
Lee, O. and Shin, D. W. (2005). On stationarity and β-mixing property of certain nonlinear GARCH(p,q,) models. Statist. Prob. Lett. 73, 25–35.Google Scholar
Li, C. W. and Li, W. K. (1996). On a double-threshold autoregressive heteroscedastic time series model. J. Appl. Econom. 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.Google Scholar
Ling, S. (2004). Estimation and testing stationarity for double-autoregressive models. J. R. Statist. Soc. B 66, 63–78.CrossRefGoogle 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. (1998). On the geometric ergodicity of a non-linear autoregressive model with an autoregressive conditional heteroscedastic term. Statist. Sinica 8, 1205–1217.Google Scholar
Lu, Z. and Jiang, Z. (2001). L 1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statist. Prob. Lett. 51, 121–130.Google Scholar
Meitz, M. and Saikkonen, P. (2006). Stability of nonlinear AR-GARCH models. Preprint, University of Helsinki.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Rabemanajara, R. and Zakoïan, J. M. (1993). Threshold ARCH models and asymmetries in volatility. J. Appl. Econometrics 8, 31–49.Google Scholar
Steinsaltz, D. (1999). Locally contractive iterative functions. Ann. Prob. 27, 1952–1952.CrossRefGoogle Scholar
Tong, H. (1981). A note on a Markov bilinear stochastic process in discrete time. J. Time Ser. Anal. 2, 279–284.Google Scholar
Tong, H. (1983). Threshold Models in Nonlinear Time Series Analysis (Notes in Stat. 21). Springer, Heidelberg.Google Scholar
Tong, H. (1990). Non-linear Time Series Analysis: A Dynamical System Approach. Oxford University Press.Google Scholar
Tong, H. and Lim, K. S. (1980). Threshold autregression, limit cycles and cyclical data (with discussion). J. R. Statist. Soc. Ser. B 42, 245–292.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar
Wu, S. and Chen, R. (2007). Threshold variable selection and threshold variable driven switching autoregressive models. Statist. Sinica 17, 241–264.Google Scholar
Zhang, Z. and Tong, H. (2004). A note on stochastic difference equations and its application to GARCH models. Chinese J. Appl. Prob. 20, 259–269.Google Scholar