The Annals of Applied Probability

Stationarity and geometric ergodicity of a class of nonlinear ARCH models

Youssef Saïdi and Jean-Michel Zakoïan

Full-text: Open access


A class of nonlinear ARCH processes is introduced and studied. The existence of a strictly stationary and β-mixing solution is established under a mild assumption on the density of the underlying independent process. We give sufficient conditions for the existence of moments. The analysis relies on Markov chain theory. The model generalizes some important features of standard ARCH models and is amenable to further analysis.

Article information

Ann. Appl. Probab. Volume 16, Number 4 (2006), 2256-2271.

First available in Project Euclid: 17 January 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 91B84: Economic time series analysis [See also 62M10]

β-mixing ergodicity GARCH-type models Markov chains nonlinear time series threshold models


Saïdi, Youssef; Zakoïan, Jean-Michel. Stationarity and geometric ergodicity of a class of nonlinear ARCH models. Ann. Appl. Probab. 16 (2006), no. 4, 2256--2271. doi:10.1214/105051606000000565.

Export citation


  • Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95--115.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307--327.
  • Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714--1730.
  • Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115--127.
  • Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17--39.
  • Cline, D. B. H. and Pu, H. H. (1998). Verifying irreducibility and continuity of a nonlinear time series. Statist. Probab. Lett. 40 139--148.
  • Cline, D. B. H. and Pu, H. H. (2004). Stability and the Lyapounov exponent of threshold AR--ARCH models. Ann. Appl. Probab. 14 1920--1949.
  • Davydov, Y. (1973). Mixing conditions for Markov chains. Theory Probab. Appl. 18 313--328.
  • Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50 987--1007.
  • Francq, C. and Zakoïan, J.-M. (2006). Mixing properties of a general class of GARCH(1, 1) models without moment assumptions on the observed process. Econometric Theory 22 815--834.
  • Hwang, S. Y. and Kim, T. Y. (2004). Power transformation and threshold modeling for ARCH innovations with applications to tests for ARCH structure. Stochastic Process. Appl. 110 295--314.
  • Ling, S. and McAleer, M. (2002). Stationarity and the existence of moments of a family of GARCH processes. J. Econometrics 106 109--117.
  • Meyn, S. P. and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability, 3rd ed. Springer, London.
  • Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6 318--334.
  • Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347--370.
  • Petruccelli, J. D. and Woolford, S. W. (1984). A threshold AR(1) model. J. Appl. Probab. 21 270--286.
  • Saïdi, Y. (2003). Etude probabiliste et statistique de modèles conditionnellement hétéroscédastiques non linéaires. Unpublished thesis, Lille 3 Univ. Available at
  • Tjøstheim, D. (1990). Non-linear time series and Markov chains. Adv. in Appl. Probab. 22 587--611.
  • Tong, H. and Lim, K. S. (1980). Threshold autoregression, limit cycles and cyclical data. J. Roy. Statist. Soc. Ser. B 42 245--292.
  • Zakoïan, J.-M. (1994). Threshold heteroskedastic models. J. Econom. Dynam. Control 18 931--955.