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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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Meitz, Mika
Saikkonen, Pentti

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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 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 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.

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Publisher Info
Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 24 (2008)
Issue (Month): 05 (October)
Pages: 1291-1320
Download reference. The following formats are available: HTML (with abstract), plain text (with abstract), BibTeX, RIS (EndNote, RefMan, ProCite), ReDIF
Handle: RePEc:cup:etheor:v:24:y:2008:i:05:p:1291-1320_08

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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. 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)
  2. 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)
    Other versions:
  3. 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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  4. 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!]
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  5. 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!]
  6. 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)
  7. 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!]
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  8. Meitz, Mika & Teräsvirta, Timo, 2004. "Evaluating models of autoregressive conditional duration," Working Paper Series in Economics and Finance 557, Stockholm School of Economics, revised 13 Dec 2004. [Downloadable!]
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  9. 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!]
  10. 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.
  11. 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)
  12. Lawrence R. Glosten & Ravi Jagannathan & David E. Runkle, 1993. "On the relation between the expected value and the volatility of the nominal excess return on stocks," Staff Report 157, Federal Reserve Bank of Minneapolis. [Downloadable!]
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  13. 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!]
  14. 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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  15. 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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  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!]
  21. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July. [Downloadable!] (restricted)
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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. Mika Meitz & Pentti Saikkonen, 2008. "Parameter Estimation in Nonlinear AR-GARCH Models," Economics Working Papers ECO2008/25, European University Institute. [Downloadable!]
    Other versions:
  3. Meitz, Mika & Saikkonen, Pentti, 2006. "Stability of nonlinear AR-GARCH models," Working Paper Series in Economics and Finance 632, Stockholm School of Economics. [Downloadable!]
    Other versions:
  4. Christina Amado & Timo Teräsvirta, 2008. "Modelling Conditional and Unconditional Heteroskedasticity with Smoothly Time-Varying Structure," CREATES Research Papers 2008-08, School of Economics and Management, University of Aarhus. [Downloadable!]
    Other versions:
  5. 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:
  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. Konstantinos Fokianos & Anders Rahbek & Dag Tjøstheim, 2009. "Poisson Autoregression," CREATES Research Papers 2009-12, School of Economics and Management, University of Aarhus. [Downloadable!]
  8. 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:
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