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Stability of nonlinear AR-GARCH models

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  • Meitz, Mika

    ()
    (Dept. of Economic Statistics, Stockholm School of Economics)

  • Saikkonen, Pentti

    ()
    (Dept. of Mathematics and Statistics, University of Helsinki)

Abstract

This paper studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and beta-mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance.

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Bibliographic Info

Paper provided by Stockholm School of Economics in its series Working Paper Series in Economics and Finance with number 632.

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Length: 23 pages
Date of creation: 01 Jun 2006
Date of revision:
Publication status: Published in Journal of Time Series Analysis, 2008, pages 453-475.
Handle: RePEc:hhs:hastef:0632

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  1. Mika Meitz & Pentti Saikkonen, 2007. "Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models," Economics Series Working Papers, University of Oxford, Department of Economics 327, University of Oxford, Department of Economics.
  2. 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, Federal Reserve Bank of Minneapolis 157, Federal Reserve Bank of Minneapolis.
  3. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  4. van Dijk, D.J.C. & Terasvirta, T. & Franses, Ph.H.B.F., 2000. "Smooth transition autoregressive models - A survey of recent developments," Econometric Institute Research Papers, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute EI 2000-23/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  5. Lundbergh, Stefan & Terasvirta, Timo, 2002. "Evaluating GARCH models," Journal of Econometrics, Elsevier, Elsevier, vol. 110(2), pages 417-435, October.
  6. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 18(01), pages 17-39, February.
  7. González-Rivera Gloria, 1998. "Smooth-Transition GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, De Gruyter, vol. 3(2), pages 1-20, July.
  8. An, H. Z. & Chen, S. G., 1997. "A note on the ergodicity of non-linear autoregressive model," Statistics & Probability Letters, Elsevier, Elsevier, vol. 34(4), pages 365-372, June.
  9. Lee, Chanho, 1998. "Asymptotics of a class of pth-order nonlinear autoregressive processes," Statistics & Probability Letters, Elsevier, Elsevier, vol. 40(2), pages 171-177, September.
  10. Saikkonen, Pentti, 2005. "Stability results for nonlinear error correction models," Journal of Econometrics, Elsevier, Elsevier, vol. 127(1), pages 69-81, July.
  11. Cline, Daren B. H. & Pu, Huay-min H., 1998. "Verifying irreducibility and continuity of a nonlinear time series," Statistics & Probability Letters, Elsevier, Elsevier, vol. 40(2), pages 139-148, September.
  12. Meitz, Mika & Saikkonen, Pentti, 2006. "Stability of nonlinear AR-GARCH models," Working Paper Series in Economics and Finance, Stockholm School of Economics 632, Stockholm School of Economics.
  13. Lu, Zudi & Jiang, Zhenyu, 2001. "L1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term," Statistics & Probability Letters, Elsevier, Elsevier, vol. 51(2), pages 121-130, January.
  14. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 99(1), pages 95-115, May.
  15. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 19(02), pages 280-310, April.
  16. Bhattacharya, Rabi & Lee, Chanho, 1995. "On geometric ergodicity of nonlinear autoregressive models," Statistics & Probability Letters, Elsevier, Elsevier, vol. 22(4), pages 311-315, March.
  17. Rabemananjara, R & Zakoian, J M, 1993. "Threshold Arch Models and Asymmetries in Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., John Wiley & Sons, Ltd., vol. 8(1), pages 31-49, Jan.-Marc.
  18. Masry, Elias & Tjøstheim, Dag, 1995. "Nonparametric Estimation and Identification of Nonlinear ARCH Time Series Strong Convergence and Asymptotic Normality: Strong Convergence and Asymptotic Normality," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 11(02), pages 258-289, February.
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Citations

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Cited by:
  1. Mika Meitz & Pentti Saikkonen, 2007. "Stability of nonlinear AR-GARCH models," Economics Series Working Papers, University of Oxford, Department of Economics 328, University of Oxford, Department of Economics.
  2. Chou, Ray Yeutien & Cai, Yijie, 2009. "Range-based multivariate volatility model with double smooth transition in conditional correlation," Global Finance Journal, Elsevier, vol. 20(2), pages 137-152.
  3. Emma M. Iglesias & Oliver Linton, 2009. "Estimation of tail thickness parameters from GJR-GARCH models," Economics Working Papers we094726, Universidad Carlos III, Departamento de Economía.
  4. Mika Meitz & Pentti Saikkonen, 2008. "Parameter estimation in nonlinear AR-GARCH models," CREATES Research Papers 2008-30, School of Economics and Management, University of Aarhus.
  5. Mika Meitz & Pentti Saikkonen, 2010. "A note on the geometric ergodicity of a nonlinear AR–ARCH model," Koç University-TUSIAD Economic Research Forum Working Papers, Koc University-TUSIAD Economic Research Forum 1003, Koc University-TUSIAD Economic Research Forum.
  6. Degiannakis, Stavros & Floros, Christos & Dent, Pamela, 2013. "Forecasting value-at-risk and expected shortfall using fractionally integrated models of conditional volatility: International evidence," International Review of Financial Analysis, Elsevier, Elsevier, vol. 27(C), pages 21-33.
  7. Jiti Gao & Maxwell King, 2011. "A New Test in Parametric Linear Models against Nonparametric Autoregressive Errors," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 20/11, Monash University, Department of Econometrics and Business Statistics.

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