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Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models

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Mika Meitz
Pentti Saikkonen

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Abstract

This paper studies a class of Markov models which 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 ?-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 which includes as special cases various first order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated non-linear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.

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Paper provided by University of Oxford, Department of Economics in its series Economics Series Working Papers with number 327.

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Date of creation: 2007
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Handle: RePEc:oxf:wpaper:327

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Keywords: Generalized Autoregressive Conditional Heteroskedasticity; Autoregressive Conditional Duration; GARCH-in-mean; Nonlinear Time Series Models; Geometric Erogidicity; Mixing; Strict Stationarity; Existence of Moments; Markov Models;

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Find related papers by JEL classification:
C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - General
C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Fernandes, Marcelo & Grammig, Joachim, 2002. "A Family of Autoregressive Conditional Duration Models," Economics Working Papers (Ensaios Economicos da EPGE) 440, Graduate School of Economics, Getulio Vargas Foundation (Brazil). [Downloadable!]
    Other versions:
  2. Hentschel, Ludger, 1995. "All in the family Nesting symmetric and asymmetric GARCH models," Journal of Financial Economics, Elsevier, vol. 39(1), pages 71-104, September. [Downloadable!] (restricted)
  3. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(02), pages 280-310, April. [Downloadable!]
    Other versions:
  4. Marcelo Cunha Medeiros & Alvaro Veiga, 2004. "Modelling multiple regimes in financial volatility with a flexible coefficient GARCH model," Textos para discussão 486, Department of Economics PUC-Rio (Brazil). [Downloadable!]
  5. Meitz, Mika & Terasvirta, Timo, 2006. "Evaluating Models of Autoregressive Conditional Duration," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 104-124, January. [Downloadable!] (restricted)
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  6. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
  7. Lundbergh, Stefan & Terasvirta, Timo, 2002. "Evaluating GARCH models," Journal of Econometrics, Elsevier, vol. 110(2), pages 417-435, October. [Downloadable!] (restricted)
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  8. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127. [Downloadable!] (restricted)
  9. Ding, Zhuanxin & Granger, Clive W. J., 1996. "Modeling volatility persistence of speculative returns: A new approach," Journal of Econometrics, Elsevier, vol. 73(1), pages 185-215, July. [Downloadable!] (restricted)
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  10. Meitz, Mika, 2006. "A Necessary And Sufficient Condition For The Strict Stationarity Of A Family Of Garch Processes," Econometric Theory, Cambridge University Press, vol. 22(05), pages 985-988, October. [Downloadable!]
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  11. Marcelo Fernandes & Marcelo Cunha Medeiros & Alvaro Veiga, 2006. "A (semi-)parametric functional coefficient autoregressive conditional duration model," Textos para discussão 535, Department of Economics PUC-Rio (Brazil). [Downloadable!]
  12. Comte, F. & Lieberman, O., 2003. "Asymptotic theory for multivariate GARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 61-84, January. [Downloadable!] (restricted)
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  14. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(01), pages 17-39, February. [Downloadable!]
  15. Ling, Shiqing & McAleer, Michael, 2002. "Stationarity and the existence of moments of a family of GARCH processes," Journal of Econometrics, Elsevier, vol. 106(1), pages 109-117, January. [Downloadable!] (restricted)
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  16. Lu, Zudi & Jiang, Zhenyu, 2001. "L1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 121-130, January. [Downloadable!] (restricted)
  17. Zhang, Michael Yuanjie & Russell, Jeffrey R. & Tsay, Ruey S., 2001. "A nonlinear autoregressive conditional duration model with applications to financial transaction data," Journal of Econometrics, Elsevier, vol. 104(1), pages 179-207, August. [Downloadable!] (restricted)
  18. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April. [Downloadable!] (restricted)
  19. M. Lanne & P. Saikkonen, . "Nonlinear GARCH Models for Highly Persistent Volatility," Sonderforschungsbereich 373 2002-20, Humboldt Universitaet Berlin.
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  20. Francq, Christian & Zako an, Jean-Michel, 2006. "Mixing Properties Of A General Class Of Garch(1,1) Models Without Moment Assumptions On The Observed Process," Econometric Theory, Cambridge University Press, vol. 22(05), pages 815-834, October. [Downloadable!]
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Konstantinos Fokianos & Anders Rahbek & Dag Tjøstheim, 2008. "Poisson Autoregression," Discussion Papers 08-35, University of Copenhagen. Department of Economics, revised Dec 2008. [Downloadable!]
  2. Silvennoinen, Annastiina & Teräsvirta, Timo, 2007. "Modelling Multivariate Autoregressive Conditional Heteroskedasticity with the Double Smooth Transition Conditional Correlation GARCH model," Working Paper Series in Economics and Finance 0652, Stockholm School of Economics.
    Other versions:
  3. Luc, BAUWENS & Nikolaus, HAUTSCH, 2006. "Modelling Financial High Frequency Data Using Point Processes," Discussion Papers (ECON - Département des Sciences Economiques) 2006039, Université catholique de Louvain, Département des Sciences Economiques. [Downloadable!]
    Other versions:
  4. Mika Meitz & Pentti Saikkonen, 2007. "Stability of nonlinear AR-GARCH models," Economics Series Working Papers 328, University of Oxford, Department of Economics. [Downloadable!]
    Other versions:
  5. Amado, Cristina & Teräsvirta, Timo, 2008. "Modelling Conditional and Unconditional Heteroskedasticity with Smoothly Time-Varying Structure," Working Paper Series in Economics and Finance 691, Stockholm School of Economics. [Downloadable!]
    Other versions:
  6. HAFNER, Christian M. & PREMINGER, Arie, 2006. "Asymptotic theory for a factor GARCH model," CORE Discussion Papers 2006071, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE). [Downloadable!]
    Other versions:
  7. Mika Meitz & Pentti Saikkonen, 2008. "Parameter Estimation in Nonlinear AR-GARCH Models," Economics Working Papers ECO2008/25, European University Institute. [Downloadable!]
    Other versions:
  8. Konstantinos Fokianos & Anders Rahbek & Dag Tjøstheim, 2009. "Poisson Autoregression," CREATES Research Papers 2009-12, School of Economics and Management, University of Aarhus. [Downloadable!]
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