Stationarity and geometric ergodicity of a class of nonlinear ARCH models

Saidi, Youssef and Zakoian, Jean-Michel (2006): Stationarity and geometric ergodicity of a class of nonlinear ARCH models. Published in: The Annals of Applied Probability , Vol. 4, No. 16 (2006): pp. 2256-2271.

Abstract

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: β-mixing, ergodicity, GARCH-type models, Markov chains, nonlinear time series, threshold models.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; G - Financial Economics > G1 - General Financial Markets

References

Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95–115. MR1894253

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307–327. MR0853051

Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714–1730. MR1188039

Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115–127. MR1165646

Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17–39. MR1885348

Cline, D. B. H. and Pu, H. H. (1998). Verifying irreducibility and continuity of a nonlinear time series. Statist. Probab. Lett. 40 139–148. MR1650873

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. MR2099657

Davydov, Y. (1973). Mixing conditions for Markov chains. Theory Probab. Appl. 18 313–328. MR0321183

Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50 987–1007. MR0666121

Francq, C. and Zako¨ıan, 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. MR2040970

Ling, S. and McAleer, M. (2002). Stationarity and the existence of moments of a family of GARCH processes. J. Econometrics 106 109–117. MR1875529

Meyn, S. P. and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability, 3rd ed. Springer, London. MR1287609 [14] Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6 318–334. MR1085577

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347–370. MR1097532

Petruccelli, J. D. and Woolford, S. W. (1984). A threshold AR(1) model. J. Appl. Probab. 21 270–286. MR0741130

Saidi, Y. (2003). Etude probabiliste et statistique de modèles conditionnellement hétéroscédastiques non linéaires. Unpublished thesis, Lille 3 Univ. Available at http://www.univ-lille3.fr/theses/saidi-youssef.pdf.

Tjøstheim, D. (1990). Non-linear time series and Markov chains. Adv. in Appl. Probab. 22 587–611. MR1066965

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.