Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T15:07:22.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolute regularity of semi-contractive GARCH-type processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  12 July 2019

Paul Doukhan  and
Michael H. Neumann
Paul Doukhan*
Affiliation:
Université Cergy-Pontoise
Michael H. Neumann*
Affiliation:
Friedrich-Schiller-Universität Jena
*
*Postal address: UMR 8088 Analyse, Géométrie et Modélisation, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.
**Postal address: Friedrich-Schiller-Universität Jena, Institut für Mathematik, Ernst-Abbe-Platz 2, 07743 Jena, Germany. Email address: michael.neumann@uni-jena.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.

Keywords

Absolute regularitycouplingGARCHINGARCHmixing

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 91 - 115
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adell, J. A. and Jodrá, P. (2006). Exact Kolmogorov and total variation distances between some familiar discrete distributions. J. Inequal. Appl. 2006, 64307, 8pp.CrossRefGoogle Scholar
Baraud, Y., Comte, F. and Viennet, G. (2001). Adaptive estimation in autoregression or β-mixing regression via model selection. Ann. Statist. 29 (3), 839–875.Google Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.CrossRefGoogle Scholar
Boussama, F. (1998). Ergodicité, mélangeances et estimation des modèles GARCH. Doctoral Thesis, University Paris 7.Google Scholar
Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. I. Kendrick Press, Heber City.Google Scholar
Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Csorgö, M. and Horvath, L. (1997). Limit Theorems in Change-Point Analysis. Wiley, Chichester.Google Scholar
Dedecker, J., Doukhan, P., Lang, G., León, J. R., Louhichi, S. and Prieur, C. (2007). Weak Dependence: With Examples and Applications (Lecture Notes Statist. 190). Springer, New York.CrossRefGoogle Scholar
Den Hollander, F. (2012). Probability theory: the coupling method. Lecture Notes, University of Leiden.Google Scholar
Douc, R., Doukhan, P. and Moulines, E. (2013). Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator. Stoch. Process. Appl. 123, 2620–2647.CrossRefGoogle Scholar
Doukhan, P. (1994). Mixing: Properties and Examples (Lecture Notes Statist. 84). Springer, Berlin.CrossRefGoogle Scholar
Doukhan, P., Massart, P. and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. l’IHP Probabilités et Statistiques 30 (2), 62–82.Google Scholar
Doukhan, P., Massart, P. and Rio, E. (1995). Invariance principles for absolutely regular empirical processes. Ann. Inst. H. Poincaré Prob. Statist. 31, 393–427.Google Scholar
Durrett, R. (1991). Probability: Theory and Examples. Wadsworth, Pacific Grove.Google Scholar
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Fokianos, K. (2012). Count time series models. In Time Series: Methods and Applications (Handbook Statist. 30), eds Subba Rao, T., Subba Rao, S. and Rao, C. R., Elsevier, Amsterdam, pp. 315–347.CrossRefGoogle Scholar
Fokianos, K. and Tjøstheim, D. (2011). Log-linear Poisson autoregression. J. Multivariate Anal. 102, 563–578.CrossRefGoogle Scholar
fokianos, K., Rahbek, A. and Tjøstheim, D. (2009). Poisson autoregression. J. Amer. Statist. Soc. 104, 1430–1439.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Francq, C. and Zakoïan, J.-M. (2010). GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley, Chichester.CrossRefGoogle Scholar
Franke, J. (2010). Weak dependence of functional INGARCH processes. Unpublished manuscript.Google Scholar
Glosten, L. R., Jagannathan, R. and Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. J. Finance 48, 1779–1801.CrossRefGoogle Scholar
Lambert, D. (1992). Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34, 1–14.CrossRefGoogle Scholar
Leucht, A. and Neumann, M. H. (2013). Degenerate U- and V-statistics under ergodicity: asymptotics, bootstrap and applications in statistics. Ann. Inst. Statist. Math. 65, 349–386.CrossRefGoogle Scholar
Leucht, A., Kreiss, J.-P. and Neumann, M. H. (2015). A model specification test for GARCH(1,1) processes. Scand. J. Stat. 42, 1167–1193.CrossRefGoogle Scholar
Lindner, A. M. (2009). Stationarity, mixing, distributional properties and moments of GARCH( p,q)-processes. In Handbook of Financial Time Series, eds Mikosch, T. et al., Springer, Berlin, pp. 43–69.CrossRefGoogle Scholar
Neumann, M. H. (2011). Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17, 1268–1284.CrossRefGoogle Scholar
Nummelin, E. and Tuominen, P. (1982). Geometric ergodicity of Harris recurrent Markov chain with applications to renewal theory. Z. Wahrscheinlichkeitsth. 12, 187–202.Google Scholar
Truquet, L. (2019). Local stationarity and time-inhomogeneous Markov chains. To appear in Ann. Statist..CrossRefGoogle Scholar
Tweedie, R. L. (1988). Invariant measures for Markov chains with no irreducibility assumptions. J. Appl. Prob. 25, 275–285.CrossRefGoogle Scholar
Viennet, G. (1997). Inequalities for absolutely regular sequences: application to density estimation. Prob. Theory Relat. Fields 107, 467–492.CrossRefGoogle Scholar
Woodard, D. B., Matteson, D. S. and Henderson, S. G. (2011). Stationarity of generalized autoregressive moving average models. Electron. J. Statist. 5, 800–828.CrossRefGoogle Scholar