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Parameter estimation in nonlinear AR–GARCH models

Author

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  • Mika Meitz

    (Koc University)

  • Pentti Saikkonen

    (University of Helsinki)

Abstract

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear 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. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. 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.

Suggested Citation

  • 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.
    • Handle: RePEc:koc:wpaper:1002
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    Cited by:

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    2. Li, Dong & Ling, Shiqing & Zakoïan, Jean-Michel, 2015. "Asymptotic inference in multiple-threshold double autoregressive models," Journal of Econometrics, Elsevier, vol. 189(2), pages 415-427.
    3. 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.
    4. Chorro, Christophe & Guégan, Dominique & Ielpo, Florian & Lalaharison, Hanjarivo, 2018. "Testing for leverage effects in the returns of US equities," Journal of Empirical Finance, Elsevier, vol. 48(C), pages 290-306.
    5. Meitz, Mika & Saikkonen, Pentti, 2011. "Parameter Estimation In Nonlinear Ar–Garch Models," Econometric Theory, Cambridge University Press, vol. 27(6), pages 1236-1278, December.
    6. Hill, Jonathan B. & Prokhorov, Artem, 2016. "GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference," Journal of Econometrics, Elsevier, vol. 190(1), pages 18-45.
    7. Meitz, Mika & Saikkonen, Pentti, 2013. "Maximum likelihood estimation of a noninvertible ARMA model with autoregressive conditional heteroskedasticity," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 227-255.
    8. Bjoern Schulte-Tillmann & Mawuli Segnon & Timo Wiedemann, 2023. "A comparison of high-frequency realized variance measures: Duration- vs. return-based approaches," CQE Working Papers 10523, Center for Quantitative Economics (CQE), University of Muenster.
    9. Matthias R. Fengler & Alexander Melnikov, 2018. "GARCH option pricing models with Meixner innovations," Review of Derivatives Research, Springer, vol. 21(3), pages 277-305, October.
    10. Haejune Oh & Sangyeol Lee, 2019. "Modified residual CUSUM test for location-scale time series models with heteroscedasticity," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1059-1091, October.
    11. Roy Cerqueti & Massimiliano Giacalone & Raffaele Mattera, 2020. "Skewed non-Gaussian GARCH models for cryptocurrencies volatility modelling," Papers 2004.11674, arXiv.org.
    12. Christophe Chorro & Dominique Guegan & Florian Ielpo & Hanjarivo Lalaharison, 2017. "Testing for Leverage Effects in the Returns of US Equities," Post-Print halshs-00973922, HAL.
    13. Annastiina Silvennoinen & Timo Ter�svirta, 2015. "Modeling Conditional Correlations of Asset Returns: A Smooth Transition Approach," Econometric Reviews, Taylor & Francis Journals, vol. 34(1-2), pages 174-197, February.
    14. Dong Li & Shiqing Ling & Jean-Michel Zakoian, 2013. "Asymptotic Inference in Multiple-Threshold Nonlinear Time Series Models," Working Papers 2013-51, Center for Research in Economics and Statistics.
    15. Vasiliki Christou & Konstantinos Fokianos, 2014. "Quasi-Likelihood Inference For Negative Binomial Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(1), pages 55-78, January.
    16. Sun, Mucun & Feng, Cong & Chartan, Erol Kevin & Hodge, Bri-Mathias & Zhang, Jie, 2019. "A two-step short-term probabilistic wind forecasting methodology based on predictive distribution optimization," Applied Energy, Elsevier, vol. 238(C), pages 1497-1505.
    17. Lee, Sangyeol & Oh, Haejune, 2015. "Entropy test and residual empirical process for autoregressive conditional duration models," Computational Statistics & Data Analysis, Elsevier, vol. 86(C), pages 1-12.
    18. 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.
    19. Christophe Chorro & Dominique Guegan & Florian Ielpo & Hanjarivo Lalaharison, 2014. "Testing for Leverage Effect in Financial Returns," Documents de travail du Centre d'Economie de la Sorbonne 14022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.

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    More about this item

    Keywords

    Nonlinear Autoregression; Generalized Autoregressive Conditional Heteroskedasticity; Nonlinear Time Series Models; Quasi-Maximum Likelihood Estimation; Strong Consistency; Asymptotic Normality;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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