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Stability of nonlinear AR-GARCH models

  • MEITZ, Mika
  • SAIKKONEN, Pentti

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 B-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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Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2006078.

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Date of creation: 00 Sep 2006
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Handle: RePEc:cor:louvco:2006078
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  1. Lundbergh, Stefan & Terasvirta, Timo, 2002. "Evaluating GARCH models," Journal of Econometrics, Elsevier, vol. 110(2), pages 417-435, October.
  2. Meitz, Mika & Saikkonen, Pentti, 2004. "Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models," SSE/EFI Working Paper Series in Economics and Finance 573, Stockholm School of Economics, revised 20 Apr 2007.
  3. Bhattacharya, Rabi & Lee, Chanho, 1995. "On geometric ergodicity of nonlinear autoregressive models," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 311-315, March.
  4. Saikkonen, Pentti, 2005. "Stability results for nonlinear error correction models," Journal of Econometrics, Elsevier, vol. 127(1), pages 69-81, July.
  5. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
  6. 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 EI 2000-23/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  7. MEITZ, Mika & SAIKKONEN, Pentti, 2006. "Stability of nonlinear AR-GARCH models," CORE Discussion Papers 2006078, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  8. An, H. Z. & Chen, S. G., 1997. "A note on the ergodicity of non-linear autoregressive model," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 365-372, June.
  9. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
  10. 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, vol. 11(02), pages 258-289, February.
  11. Cline, Daren B. H. & Pu, Huay-min H., 1998. "Verifying irreducibility and continuity of a nonlinear time series," Statistics & Probability Letters, Elsevier, vol. 40(2), pages 139-148, September.
  12. Lee, Chanho, 1998. "Asymptotics of a class of pth-order nonlinear autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 40(2), pages 171-177, September.
  13. 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.
  14. 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.
  15. Rabemananjara, R & Zakoian, J M, 1993. "Threshold Arch Models and Asymmetries in Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(1), pages 31-49, Jan.-Marc.
  16. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  17. 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.
  18. González-Rivera Gloria, 1998. "Smooth-Transition GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 3(2), pages 1-20, July.
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