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Variance targeting estimation of multivariate GARCH models

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  • Francq, Christian
  • Horvath, Lajos
  • Zakoian, Jean-Michel

Abstract

We establish the strong consistency and the asymptotic normality of the variance-targeting estimator (VTE) of the parameters of the multivariate CCC-GARCH($p,q$) processes. This method alleviates the numerical difficulties encountered in the maximization of the quasi likelihood by using an estimator of the unconditional variance. It is shown that the distribution of the VTE can be consistently estimated by a simple residual bootstrap technique. We also use the VTE for testing the model adequacy. A test statistic in the spirit of the score test is constructed, and its asymptotic properties are derived under the null assumption that the model is well specified. An extension of the VT method to asymmetric CCC-GARCH models incorporating leverage effects is studied. Numerical illustrations are provided and an empirical application based on daily exchange rates is proposed.

Suggested Citation

  • Francq, Christian & Horvath, Lajos & Zakoian, Jean-Michel, 2014. "Variance targeting estimation of multivariate GARCH models," MPRA Paper 57794, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:57794
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    References listed on IDEAS

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    1. Sébastien Laurent & Jeroen V. K. Rombouts & Francesco Violante, 2012. "On the forecasting accuracy of multivariate GARCH models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 27(6), pages 934-955, September.
    2. Francq, Christian & Zakoïan, Jean-Michel, 2009. "Testing the Nullity of GARCH Coefficients: Correction of the Standard Tests and Relative Efficiency Comparisons," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 313-324.
    3. Boudt, Kris & Daníelsson, Jón & Laurent, Sébastien, 2013. "Robust forecasting of dynamic conditional correlation GARCH models," International Journal of Forecasting, Elsevier, vol. 29(2), pages 244-257.
    4. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(1), pages 122-150, February.
    5. Rasmus S. Pedersen & Anders Rahbek, 2014. "Multivariate variance targeting in the BEKK–GARCH model," Econometrics Journal, Royal Economic Society, vol. 17(1), pages 24-55, February.
    6. 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(3), pages 722-729, June.
    7. Sébastien Laurent & Luc Bauwens & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 79-109.
    8. Zhu, Dongming & Zinde-Walsh, Victoria, 2009. "Properties and estimation of asymmetric exponential power distribution," Journal of Econometrics, Elsevier, vol. 148(1), pages 86-99, January.
    9. Christian Francq & Lajos Horváth, 2011. "Merits and Drawbacks of Variance Targeting in GARCH Models," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 9(4), pages 619-656.
    10. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(1), pages 70-86, February.
    11. Francq, Christian & Zakoïan, Jean-Michel, 2012. "Qml Estimation Of A Class Of Multivariate Asymmetric Garch Models," Econometric Theory, Cambridge University Press, vol. 28(1), pages 179-206, February.
    12. He, Changli & Teräsvirta, Timo, 2004. "An Extended Constant Conditional Correlation Garch Model And Its Fourth-Moment Structure," Econometric Theory, Cambridge University Press, vol. 20(5), pages 904-926, October.
    13. Kristensen, Dennis & Linton, Oliver, 2004. "03.5.2. Consistent Standard Errors for Target Variance Approach to GARCH Estimation—Solution," Econometric Theory, Cambridge University Press, vol. 20(5), pages 990-993, October.
    14. Christoffersen, Peter, 2011. "Elements of Financial Risk Management," Elsevier Monographs, Elsevier, edition 2, number 9780123744487.
    15. Igor Vaynman & Brendan K. Beare, 2014. "Stable Limit Theory for the Variance Targeting Estimator," Advances in Econometrics, in: Yoosoon Chang & Thomas B. Fomby & Joon Y. Park (ed.),Essays in Honor of Peter C. B. Phillips, volume 33, pages 639-672, Emerald Publishing Ltd.
    16. 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.
    17. Bollerslev, Tim, 1990. "Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model," The Review of Economics and Statistics, MIT Press, vol. 72(3), pages 498-505, August.
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    Cited by:

    1. Thieu, Le Quyen, 2016. "Variance targeting estimation of the BEKK-X model," MPRA Paper 75572, University Library of Munich, Germany.
    2. BAUWENS, Luc & BRAIONE, Manuela & STORTI, Giuseppe, 2016. "Multiplicative Conditional Correlation Models for Realized Covariance Matrices," CORE Discussion Papers 2016041, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Giuseppe Cavaliere & Rasmus Søndergaard Pedersen & Anders Rahbek, 2018. "The Fixed Volatility Bootstrap for a Class of Arch(q) Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 920-941, November.
    4. Francq, Christian & Zakoian, Jean-Michel, 2019. "Virtual Historical Simulation for estimating the conditional VaR of large portfolios," MPRA Paper 95965, University Library of Munich, Germany.
    5. Barassi, Marco & Horvath, Lajos & Zhao, Yuqian, 2018. "Change Point Detection in the Conditional Correlation Structure of Multivariate Volatility Models," MPRA Paper 87837, University Library of Munich, Germany.
    6. Qi Li & Fukang Zhu, 2020. "Mean targeting estimator for the integer-valued GARCH(1, 1) model," Statistical Papers, Springer, vol. 61(2), pages 659-679, April.
    7. de Almeida, Daniel & Hotta, Luiz K. & Ruiz, Esther, 2018. "MGARCH models: Trade-off between feasibility and flexibility," International Journal of Forecasting, Elsevier, vol. 34(1), pages 45-63.
    8. Lyubimov, Ivan L. (Любимов, Иван) & Kazakova, Maria V. (Казакова, Мария), 2017. "The Demand for Production Inputs as the Reflection of the Level of Property Rights Protection
      [Структура Спроса На Факторы Производства Как Отражение Защищенности Прав Собственности]
      ," Economic Policy, Russian Presidential Academy of National Economy and Public Administration, vol. 4, pages 30-59, August.
    9. Eric Beutner & Alexander Heinemann & Stephan Smeekes, 2018. "A Residual Bootstrap for Conditional Value-at-Risk," Papers 1808.09125, arXiv.org, revised Jul 2020.
    10. Francq, C. & Jiménez-Gamero, M.D. & Meintanis, S.G., 2017. "Tests for conditional ellipticity in multivariate GARCH models," Journal of Econometrics, Elsevier, vol. 196(2), pages 305-319.
    11. Monica Billio & Massimiliano Caporin & Lorenzo Frattarolo & Loriana Pelizzon, 2016. "Networks in risk spillovers: a multivariate GARCH perspective," Working Papers 2016:03, Department of Economics, University of Venice "Ca' Foscari".

    More about this item

    Keywords

    Adequacy Test for CCC-GARCH models; Bootstrap; Leverage Effect; Quasi Maximum Likelihood Estimation; Variance Targeting Estimator;

    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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