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Stability of nonlinear AR-GARCH models

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  • MEITZ, Mika
  • SAIKKONEN, Pentti

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 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. 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.
  2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  3. 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.
  4. González-Rivera Gloria, 1998. "Smooth-Transition GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 3(2), pages 1-20, July.
  5. Bhattacharya, Rabi & Lee, Chanho, 1995. "On geometric ergodicity of nonlinear autoregressive models," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 311-315, March.
  6. Lundbergh, Stefan & Terasvirta, Timo, 2002. "Evaluating GARCH models," Journal of Econometrics, Elsevier, vol. 110(2), pages 417-435, October.
  7. 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.
  8. 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.
  9. Shiqing Ling & Michael McAleer, 2001. "Asymptotic Theory for a Vector ARMA-GARCH Model," ISER Discussion Paper 0549, Institute of Social and Economic Research, Osaka University.
  10. Meitz, Mika & Saikkonen, Pentti, 2004. "Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models," Working Paper Series in Economics and Finance 573, Stockholm School of Economics, revised 20 Apr 2007.
  11. 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.
  12. 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).
  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.
  14. Lee, Chanho, 1998. "Asymptotics of a class of pth-order nonlinear autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 40(2), pages 171-177, September.
  15. 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.
  16. 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.
  17. 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.
  18. Saikkonen, Pentti, 2005. "Stability results for nonlinear error correction models," Journal of Econometrics, Elsevier, vol. 127(1), pages 69-81, July.
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Cited by:
  1. Mika Meitz & Pentti Saikkonen, 2010. "Parameter estimation in nonlinear AR–GARCH models," Koç University-TUSIAD Economic Research Forum Working Papers 1002, Koc University-TUSIAD Economic Research Forum.
  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, 2007. "Stability of nonlinear AR-GARCH models," Economics Series Working Papers 328, University of Oxford, Department of Economics.
  5. 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, vol. 27(C), pages 21-33.
  6. Jiti Gao & Maxwell King, 2011. "A New Test in Parametric Linear Models against Nonparametric Autoregressive Errors," Monash Econometrics and Business Statistics Working Papers 20/11, Monash University, Department of Econometrics and Business Statistics.
  7. 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 1003, Koc University-TUSIAD Economic Research Forum.

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