Stationarity and geometric ergodicity of a class of nonlinear ARCH models
Youssef Saïdi and Jean-Michel Zakoïan
Source: Ann. Appl. Probab. Volume 16, Number 4 (2006), 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.
Primary Subjects: 60G10, 60J05
Secondary Subjects: 62M10, 91B84
Keywords: β-mixing; ergodicity; GARCH-type models; Markov chains; nonlinear time series; threshold models
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
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1169065224
Digital Object Identifier: doi:10.1214/105051606000000565
References
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95--115.
Mathematical Reviews (MathSciNet):
MR1894253
Digital Object Identifier: doi:10.1016/S0304-4149(01)00156-9
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307--327.
Mathematical Reviews (MathSciNet):
MR0853051
Digital Object Identifier: doi:10.1016/0304-4076(86)90063-1
Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714--1730.
Mathematical Reviews (MathSciNet):
MR1188039
Digital Object Identifier: doi:10.1214/aop/1176989526
Project Euclid: euclid.aop/1176989526
Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115--127.
Mathematical Reviews (MathSciNet):
MR1165646
Digital Object Identifier: doi:10.1016/0304-4076(92)90067-2
Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17--39.
Mathematical Reviews (MathSciNet):
MR1885348
Digital Object Identifier: doi:10.1017/S0266466602181023
JSTOR: links.jstor.org
Cline, D. B. H. and Pu, H. H. (1998). Verifying irreducibility and continuity of a nonlinear time series. Statist. Probab. Lett. 40 139--148.
Mathematical Reviews (MathSciNet):
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.
Mathematical Reviews (MathSciNet):
MR2099657
Digital Object Identifier: doi:10.1214/105051604000000431
Project Euclid: euclid.aoap/1099674083
Davydov, Y. (1973). Mixing conditions for Markov chains. Theory Probab. Appl. 18 313--328.
Mathematical Reviews (MathSciNet):
MR0321183
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50 987--1007.
Mathematical Reviews (MathSciNet):
MR0666121
Digital Object Identifier: doi:10.2307/1912773
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR2291218
Digital Object Identifier: doi:10.1017/S0266466606060373
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.
Mathematical Reviews (MathSciNet):
MR2040970
Digital Object Identifier: doi:10.1016/j.spa.2003.11.001
Ling, S. and McAleer, M. (2002). Stationarity and the existence of moments of a family of GARCH processes. J. Econometrics 106 109--117.
Mathematical Reviews (MathSciNet):
MR1875529
Digital Object Identifier: doi:10.1016/S0304-4076(01)00090-2
Meyn, S. P. and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability, 3rd ed. Springer, London.
Mathematical Reviews (MathSciNet):
MR1287609
Zentralblatt MATH:
0925.60001
Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6 318--334.
Mathematical Reviews (MathSciNet):
MR1085577
JSTOR: links.jstor.org
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347--370.
Mathematical Reviews (MathSciNet):
MR1097532
Digital Object Identifier: doi:10.2307/2938260
JSTOR: links.jstor.org
Petruccelli, J. D. and Woolford, S. W. (1984). A threshold AR(1) model. J. Appl. Probab. 21 270--286.
Mathematical Reviews (MathSciNet):
MR0741130
Digital Object Identifier: doi:10.2307/3213639
JSTOR: links.jstor.org
Saïdi, 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.
Mathematical Reviews (MathSciNet):
MR1066965
Digital Object Identifier: doi:10.2307/1427459
JSTOR: links.jstor.org
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.
The Annals of Applied Probability
