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Optimal predictions of powers of conditionally heteroskedastic processes

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  • Francq, Christian
  • Zakoian, Jean-Michel

Abstract

In conditionally heteroskedastic models, the optimal prediction of powers, or logarithms, of the absolute process has a simple expression in terms of the volatility process and an expectation involving the independent process. A standard procedure for estimating this prediction is to estimate the volatility by gaussian quasi-maximum likelihood (QML) in a first step, and to use empirical means based on rescaled innovations to estimate the expectation in a second step. This paper proposes an alternative one-step procedure, based on an appropriate non-gaussian QML estimation of the model, and establishes the asymptotic properties of the two approaches. Their performances are compared for finite-order GARCH models and for the infinite ARCH. For the standard GARCH(p, q) and the Asymmetric Power GARCH(p,q), it is shown that the ARE of the estimators only depends on the prediction problem and some moments of the independent process. An application to indexes of major stock exchanges is proposed.

Suggested Citation

  • Francq, Christian & Zakoian, Jean-Michel, 2010. "Optimal predictions of powers of conditionally heteroskedastic processes," MPRA Paper 22155, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:22155
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    Citations

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    Cited by:

    1. Ziel, Florian, 2016. "Iteratively reweighted adaptive lasso for conditional heteroscedastic time series with applications to AR–ARCH type processes," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 773-793.
    2. Wang, Guochang & Zhu, Ke & Li, Guodong & Li, Wai Keung, 2022. "Hybrid quantile estimation for asymmetric power GARCH models," Journal of Econometrics, Elsevier, vol. 227(1), pages 264-284.
    3. Chen, Min & Zhu, Ke, 2013. "Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations," MPRA Paper 50487, University Library of Munich, Germany.
    4. Francq, Christian & Zakoïan, Jean-Michel, 2015. "Risk-parameter estimation in volatility models," Journal of Econometrics, Elsevier, vol. 184(1), pages 158-173.
    5. Li, Dong & Ling, Shiqing & Zhu, Ke, 2016. "ZD-GARCH model: a new way to study heteroscedasticity," MPRA Paper 68621, University Library of Munich, Germany.
    6. Christian Francq & Lajos Horváth & Jean-Michel Zakoïan, 2016. "Variance Targeting Estimation of Multivariate GARCH Models," Journal of Financial Econometrics, Oxford University Press, vol. 14(2), pages 353-382.
    7. Mohamed El Ghourabi & Christian Francq & Fedya Telmoudi, 2016. "Consistent Estimation of the Value at Risk When the Error Distribution of the Volatility Model is Misspecified," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(1), pages 46-76, January.
    8. Florian Ziel, 2015. "Iteratively reweighted adaptive lasso for conditional heteroscedastic time series with applications to AR-ARCH type processes," Papers 1502.06557, arXiv.org, revised Dec 2015.
    9. Francq, Christian & Zakoian, Jean-Michel, 2015. "Looking for efficient qml estimation of conditional value-at-risk at multiple risk levels," MPRA Paper 67195, University Library of Munich, Germany.
    10. Francq, Christian & Zakoïan, Jean-Michel, 2020. "Virtual Historical Simulation for estimating the conditional VaR of large portfolios," Journal of Econometrics, Elsevier, vol. 217(2), pages 356-380.
    11. Francq, Christian & Zakoïan, Jean-Michel, 2022. "Testing the existence of moments for GARCH processes," Journal of Econometrics, Elsevier, vol. 227(1), pages 47-64.
    12. Carnero M. Angeles & Pérez Ana, 2021. "Outliers and misleading leverage effect in asymmetric GARCH-type models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 25(1), pages 1-19, February.
    13. Zhu, Ke & Li, Wai Keung, 2013. "A new Pearson-type QMLE for conditionally heteroskedastic models," MPRA Paper 52344, University Library of Munich, Germany.
    14. Christian Francq & Jean-Michel Zakoian, 2014. "Multi-level Conditional VaR Estimation in Dynamic Models," Working Papers 2014-01, Center for Research in Economics and Statistics.
    15. Guochang Wang & Ke Zhu & Guodong Li & Wai Keung Li, 2019. "Hybrid quantile estimation for asymmetric power GARCH models," Papers 1911.09343, arXiv.org.
    16. Zhu, Ke, 2015. "Hausman tests for the error distribution in conditionally heteroskedastic models," MPRA Paper 66991, University Library of Munich, Germany.
    17. Chen, Min & Zhu, Ke, 2015. "Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations," Journal of Econometrics, Elsevier, vol. 189(2), pages 313-320.
    18. Aknouche, Abdelhakim, 2015. "Unified quasi-maximum likelihood estimation theory for stable and unstable Markov bilinear processes," MPRA Paper 69572, University Library of Munich, Germany.
    19. Aknouche, Abdelhakim & Touche, Nassim, 2015. "Weighted least squares-based inference for stable and unstable threshold power ARCH processes," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 108-115.
    20. Loïc Cantin & Christian Francq & Jean-Michel Zakoïan, 2022. "Estimating dynamic systemic risk measures," Working Papers 2022-11, Center for Research in Economics and Statistics.
    21. Francq, Christian & Lepage, Guillaume & Zakoïan, Jean-Michel, 2011. "Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE," Journal of Econometrics, Elsevier, vol. 165(2), pages 246-257.
    22. Li, Dong & Zhang, Xingfa & Zhu, Ke & Ling, Shiqing, 2018. "The ZD-GARCH model: A new way to study heteroscedasticity," Journal of Econometrics, Elsevier, vol. 202(1), pages 1-17.
    23. Aknouche, Abdelhakim & Demmouche, Nacer & Touche, Nassim, 2018. "Bayesian MCMC analysis of periodic asymmetric power GARCH models," MPRA Paper 91136, University Library of Munich, Germany.
    24. Aknouche, Abdelhakim & Al-Eid, Eid & Demouche, Nacer, 2016. "Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models," MPRA Paper 75770, University Library of Munich, Germany, revised 19 Dec 2016.
    25. Abdelhakim Aknouche & Eid Al-Eid & Nacer Demouche, 2018. "Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 485-511, October.

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    More about this item

    Keywords

    APARCH; Infinite ARCH; Conditional Heteroskedasticity; Efficiency of estimators; GARCH; Prediction; Quasi Maximum Likelihood Estimation;
    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
    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics

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