Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-26T03:38:59.987Z Has data issue: false hasContentIssue false
  • English
  • Français

ERGODICITY, MIXING, AND EXISTENCE OF MOMENTS OF A CLASS OF MARKOV MODELS WITH APPLICATIONS TO GARCH AND ACD MODELS

Published online by Cambridge University Press:  11 June 2008

Mika Meitz  and
Pentti Saikkonen
Mika Meitz*
Affiliation:
University of Oxford
Pentti Saikkonen*
Affiliation:
University of Helsinki
*
Address correspondence to Mika Meitz, Department of Economics, University of Oxford, Manor Road Building, Manor Road, Oxford, OX1 3UQ, United Kingdom; e-mail: [email protected].
Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 University of Helsinki, Finland; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies a class of Markov models that consist of two components. Typically, one of the components is observable and the other is unobservable or “hidden.” Conditions under which geometric ergodicity of the unobservable component is inherited by the joint process formed of the two components are given. This implies existence of initial values such that the joint process is strictly stationary and β-mixing. In addition to this, conditions for the existence of moments are also obtained, and extensions to the case of nonstationary initial values are provided. All these results are applied to a general model that includes as special cases various first-order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated nonlinear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 5 , October 2008 , pp. 1291 - 1320
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Basrak, B., Davis, R.A., & Mikosch, T. (2002) Regular variation of GARCH processes. Stochastic Processes and Their Applications 99, 95–115.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992) Stationarity of GARCH processes and some nonnegative time series. Journal of Econometrics 52, 115–127.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2005) Erratum: Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models. Manuscript, University of Rochester and New York University.Google Scholar
Chan, K.S. (1990) Deterministic stability, stochastic stability, and ergodicity. Appendix in H. Tong, Non-linear Time Series: A Dynamical System Approach, pp. 448–466. Clarendon Press.Google Scholar
Cline, D.B.H. (2007) Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models. Advances in Applied Probability 39, 462–491.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Ding, Z. & Granger, C.W.J. (1996) Modeling volatility persistence of speculative returns: A new approach. Journal of Econometrics 73, 185–215.CrossRefGoogle Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Springer-Verlag.CrossRefGoogle Scholar
Duan, J.-C. (1997) Augmented GARCH(p,q) process and its diffusion limit. Journal of Econometrics 79, 97–127.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Engle, R.F. & Russell, J.R. (1998) Autoregressive conditional duration: A new model for irregularly spaced transaction data. Econometrica 66, 1127–1162.CrossRefGoogle Scholar
Fernandes, M. & Grammig, J. (2006) A family of autoregressive conditional duration models. Journal of Econometrics 130, 1–23.CrossRefGoogle Scholar
Fernandes, M., Medeiros, M.C., & Veiga, A. (2006) A (Semi-)parametric Functional Coefficient Autoregressive Conditional Duration Model. Textos para discussão 535, Department of Economics, PUC-Rio.Google Scholar
Francq, C. & Zakoïan, J.-M. (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Genon-Catalot, V., Jeantheau, T., & Larédo, C. (2000) Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6, 1051–1079.CrossRefGoogle Scholar
Glosten, L.W., Jaganathan, R., & Runkle, D.E. (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
González-Rivera, G. (1998) Smooth-transition GARCH models. Studies in Nonlinear Dynamics and Econometrics 3, 61–78.Google Scholar
Hagerud, G.E. (1996) A Smooth Transition ARCH Model for Asset Returns. Working Paper Series in Economics and Finance 162, Stockholm School of Economics.Google Scholar
Hentschel, L. (1995) All in the family: Nesting symmetric and asymmetric GARCH models. Journal of Financial Economics 39, 71–104.CrossRefGoogle Scholar
Kristensen, D. (2007) Uniform Ergodicity of a Class of Markov Chains with Applications to Time Series Models. Manuscript, Columbia University.Google Scholar
Lanne, M. & Saikkonen, P. (2005) Nonlinear GARCH models for highly persistent volatility. Econometrics Journal 8, 251–276.CrossRefGoogle Scholar
Lee, O. & Shin, D.W. (2005) On stationarity and β-mixing property of certain nonlinear GARCH(p,q) models. Statistics and Probability Letters 73, 25–35.CrossRefGoogle Scholar
Liebscher, E. (2005) Towards a unified approach for proving geometric ergodicity and mixing properties of nonlinear autoregressive processes. Journal of Time Series Analysis 26, 669–689.CrossRefGoogle Scholar
Ling, S. (1999) On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. Journal of Applied Probability 36, 688–705.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2002) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106, 109–117.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2003) Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280–310.CrossRefGoogle Scholar
Lu, Z. (1998) On the geometric ergodicity of a non-linear autoregressive model with an autoregressive conditional heteroscedastic term. Statistica Sinica 8, 1205–1217.Google Scholar
Lu, Z. & Jiang, Z. (2001) L 1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statistics and Probability Letters 51, 121–130.CrossRefGoogle Scholar
Lundbergh, S. & Teräsvirta, T. (2002) Evaluating GARCH models. Journal of Econometrics 110, 417–435.CrossRefGoogle Scholar
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 11, 258–289.CrossRefGoogle Scholar
Medeiros, M.C. & Veiga, A. (2006) Modeling multiple regimes in financial volatility with a flexible coefficient GARCH(1,1) model, Textos para discussão 486, Department of Economics, PUC-Rio.Google Scholar
Meitz, M. (2006) A necessary and sufficient condition for the strict stationarity of a family of GARCH processes. Econometric Theory 22, 985–988.CrossRefGoogle Scholar
Meitz, M. & Teräsvirta, T. (2006) Evaluating models of autoregressive conditional duration. Journal of Business & Economic Statistics 24, 104–124.CrossRefGoogle Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.CrossRefGoogle Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.CrossRefGoogle Scholar
Zhang, M.Y., Russell, J.R., & Tsay, R.S. (2001) A nonlinear autoregressive conditional duration model with applications to financial transaction data. Journal of Econometrics 104, 179–207.CrossRefGoogle Scholar