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Parameter estimation in nonlinear AR-GARCH models

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  • Mika Meitz
  • Pentti Saikkonen

    ()
    (School of Economics and Management, University of Aarhus, Denmark and CREATES)

Abstract

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a functional coefficient autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.

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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2008-30.

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Length: 58
Date of creation: 09 Jun 2008
Date of revision:
Handle: RePEc:aah:create:2008-30

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Web page: http://www.econ.au.dk/afn/

Related research

Keywords: AR-GARCH; asymptotic normality; consistency; nonlinear time series; quasi maximum likelihood estimation;

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References

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  1. Escanciano, Juan Carlos, 2009. "Quasi-Maximum Likelihood Estimation Of Semi-Strong Garch Models," Econometric Theory, Cambridge University Press, vol. 25(02), pages 561-570, April.
  2. 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.
  3. González-Rivera Gloria, 1998. "Smooth-Transition GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 3(2), pages 1-20, July.
  4. 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 327, University of Oxford, Department of Economics.
  5. Theis Lange & Anders Rahbek & S�ren Tolver Jensen, 2011. "Estimation and Asymptotic Inference in the AR-ARCH Model," Econometric Reviews, Taylor & Francis Journals, vol. 30(2), pages 129-153.
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  8. 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).
  9. 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.
  10. Jensen, S ren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(06), pages 1203-1226, December.
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  14. 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.
  15. 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.
  16. Kristensen, Dennis & Rahbek, Anders, 2005. "ASYMPTOTICS OF THE QMLE FOR A CLASS OF ARCH(q) MODELS," Econometric Theory, Cambridge University Press, vol. 21(05), pages 946-961, October.
  17. 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.
  18. Francq, Christian & Zakoian, Jean-Michel, 2007. "Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1265-1284, September.
  19. 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.
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  23. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  24. Ling, Shiqing, 2007. "Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models," Journal of Econometrics, Elsevier, vol. 140(2), pages 849-873, October.
  25. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
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Citations

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Cited by:
  1. Cristina Amado & Timo Teräsvirta, 2011. "Modelling Volatility by Variance Decomposition," NIPE Working Papers 01/2011, NIPE - Universidade do Minho.
  2. KIlIç, Rehim, 2011. "Long memory and nonlinearity in conditional variances: A smooth transition FIGARCH model," Journal of Empirical Finance, Elsevier, vol. 18(2), pages 368-378, March.
  3. Meitz, Mika & Saikkonen, Pentti, 2011. "Parameter Estimation In Nonlinear Ar–Garch Models," Econometric Theory, Cambridge University Press, vol. 27(06), pages 1236-1278, December.
  4. Dong Li & Shiqing Ling & Jean-Michel Zakoian, 2013. "Asymptotic Inference in Multiple-Threshold Nonlinear Time Series Models," Working Papers 2013-51, Centre de Recherche en Economie et Statistique.
  5. Lambert, Philippe & Laurent, Sébastien & Veredas, David, 2012. "Testing conditional asymmetry: A residual-based approach," Journal of Economic Dynamics and Control, Elsevier, vol. 36(8), pages 1229-1247.
  6. Mika Meitz & Pentti Saikkonen, 2012. "Maximum Likelihood Estimation of a Noninvertible ARMA Model with Autoregressive Conditional Heteroskedasticity," Koç University-TUSIAD Economic Research Forum Working Papers 1226, Koc University-TUSIAD Economic Research Forum.
  7. Annastiina Silvennoinen & Timo Teräsvirta, 2012. "Modelling conditional correlations of asset returns: A smooth transition approach," CREATES Research Papers 2012-09, School of Economics and Management, University of Aarhus.

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