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Structure and Asymptotic Theory for Nonlinear Models with GARCH Errors

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  • Felix Chan
  • Michael McAleer

    ()
    (University of Canterbury)

  • Marcelo C. Medeiros

Abstract

Nonlinear time series models, especially those with regime-switching and conditionally heteroskedastic errors, have become increasingly popular in the economics and finance literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the processes or the associated asymptotic theory. In this paper, we first derive necessary conditions for strict stationarity and ergodicity of three different specifications of the first-order smooth transition autoregressions with heteroskedastic errors. This is important, among other reasons, to establish the conditions under which the traditional LMlinearity tests based on Taylor expansions are valid. Second, we provide sufficient conditions for consistency and asymptotic normality of the Quasi- Maximum Likelihood Estimator for a general nonlinear conditional mean model with first-order GARCH errors.

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File URL: http://www.econ.canterbury.ac.nz/RePEc/cbt/econwp/1079.pdf
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Bibliographic Info

Paper provided by University of Canterbury, Department of Economics and Finance in its series Working Papers in Economics with number 10/79.

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Length: 28 pages
Date of creation: 01 Dec 2010
Date of revision:
Handle: RePEc:cbt:econwp:10/79

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Keywords: Nonlinear time series; regime-switching; smooth transition; STAR; GARCH; log-moment; moment conditions; asymptotic theory;

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  1. He, Changli & Terasvirta, Timo, 1999. "Properties of moments of a family of GARCH processes," Journal of Econometrics, Elsevier, vol. 92(1), pages 173-192, September.
  2. Hwang, S. Y. & Woo, Mi-Ja, 2001. "Threshold ARCH(1) processes: asymptotic inference," Statistics & Probability Letters, Elsevier, vol. 53(1), pages 11-20, May.
  3. Lee, Chanho, 1998. "Asymptotics of a class of pth-order nonlinear autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 40(2), pages 171-177, September.
  4. 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.
  5. Ling, Shiqing & McAleer, Michael, 2002. "NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS," Econometric Theory, Cambridge University Press, vol. 18(03), pages 722-729, June.
  6. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  7. 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.
  8. McAleer, Michael & Chan, Felix & Marinova, Dora, 2007. "An econometric analysis of asymmetric volatility: Theory and application to patents," Journal of Econometrics, Elsevier, vol. 139(2), pages 259-284, August.
  9. 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.
  10. 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.
  11. Comte, F. & Lieberman, O., 2003. "Asymptotic theory for multivariate GARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 61-84, January.
  12. Felix Chan & Michael McAleer, 2002. "Maximum likelihood estimation of STAR and STAR-GARCH models: theory and Monte Carlo evidence," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 509-534.
  13. Bhattacharya, Rabi & Lee, Chanho, 1995. "On geometric ergodicity of nonlinear autoregressive models," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 311-315, March.
  14. McAleer, Michael, 2005. "Automated Inference And Learning In Modeling Financial Volatility," Econometric Theory, Cambridge University Press, vol. 21(01), pages 232-261, February.
  15. Mayte Suarez Farinãs & Carlos Eduardo Pedreira & Marcelo C. Medeiros, 2003. "Local-global neural networks: a new approach for nonlinear time series modelling," Textos para discussão 470, Department of Economics PUC-Rio (Brazil).
  16. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(01), pages 70-86, February.
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