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Estimating Multivariate Volatility Models Equation by Equation

Author

Listed:
  • Christian Francq

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, IP Paris - Institut Polytechnique de Paris)

  • Jean-Michel Zakoïan

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

Abstract

Summary The paper investigates the estimation of a wide class of multivariate volatility models. Instead of estimating an m-multivariate volatility model, a much simpler and numerically efficient method consists in estimating m univariate generalized auto-regressive conditional heteroscedasticity type models equation by equation in the first step, and a correlation matrix in the second step. Strong consistency and asymptotic normality of the equation-by-equation estimator are established in a very general framework, including dynamic conditional correlation models. The equation-by-equation estimator can be used to test the restrictions imposed by a particular multivariate generalized auto-regressive conditional heteroscedasticity specification. For general constant conditional correlation models, we obtain the consistency and asymptotic normality of the two-step estimator. Comparisons with the global method, in which the model parameters are estimated in one step, are provided. Monte Carlo experiments and applications to financial series illustrate the interest of the approach.

Suggested Citation

  • Christian Francq & Jean-Michel Zakoïan, 2016. "Estimating Multivariate Volatility Models Equation by Equation," Post-Print hal-05417324, HAL.
    • Handle: RePEc:hal:journl:hal-05417324
    DOI: 10.1111/rssb.12126
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    2. Xiaoning Kang & Xinwei Deng & Kam‐Wah Tsui & Mohsen Pourahmadi, 2020. "On variable ordination of modified Cholesky decomposition for estimating time‐varying covariance matrices," International Statistical Review, International Statistical Institute, vol. 88(3), pages 616-641, December.
    3. Geert Dhaene & Piet Sercu & Jianbin Wu, 2022. "Volatility spillovers: A sparse multivariate GARCH approach with an application to commodity markets," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(5), pages 868-887, May.
    4. Kawakatsu Hiroyuki, 2021. "Simple Multivariate Conditional Covariance Dynamics Using Hyperbolically Weighted Moving Averages," Journal of Econometric Methods, De Gruyter, vol. 10(1), pages 33-52, January.
    5. Beutner, Eric & Heinemann, Alexander & Smeekes, Stephan, 2024. "A residual bootstrap for conditional Value-at-Risk," Journal of Econometrics, Elsevier, vol. 238(2).
    6. Yuchang Lin & Qianqian Zhu, 2024. "On vector linear double autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(3), pages 376-397, May.
    7. Fokianos, Konstantinos, 2024. "Multivariate Count Time Series Modelling," Econometrics and Statistics, Elsevier, vol. 31(C), pages 100-116.
    8. Simon Hetland, 2020. "Spectral Targeting Estimation of $\lambda$-GARCH models," Papers 2007.02588, arXiv.org.
    9. Caporin, Massimiliano & Malik, Farooq, 2020. "Do structural breaks in volatility cause spurious volatility transmission?," Journal of Empirical Finance, Elsevier, vol. 55(C), pages 60-82.
    10. Boubacar Maïnassara, Y. & Kadmiri, O. & Saussereau, B., 2022. "Estimation of multivariate asymmetric power GARCH models," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    11. Veraart, Almut E.D., 2019. "Modeling, simulation and inference for multivariate time series of counts using trawl processes," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 110-129.
    12. Campos-Martins, Susana & Amado, Cristina, 2025. "Modelling dynamic interdependence in nonstationary variances with an application to carbon markets," Journal of Economic Dynamics and Control, Elsevier, vol. 173(C).
    13. Eric Beutner & Julia Schaumburg & Barend Spanjers, 2024. "Bootstrapping GARCH Models Under Dependent Innovations," Tinbergen Institute Discussion Papers 24-008/III, Tinbergen Institute.
    14. Darolles, Serge & Francq, Christian & Laurent, Sébastien, 2018. "Asymptotics of Cholesky GARCH models and time-varying conditional betas," Journal of Econometrics, Elsevier, vol. 204(2), pages 223-247.
    15. Francq, Christian & Sucarrat, Genaro, 2017. "An equation-by-equation estimator of a multivariate log-GARCH-X model of financial returns," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 16-32.
    16. Simos Meintanis & Bojana Milošević & Marko Obradović & Mirjana Veljović, 2024. "Goodness‐of‐fit tests for the multivariate Student‐t distribution based on i.i.d. data, and for GARCH observations," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(2), pages 298-319, March.
    17. 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.
    18. 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.
    19. D’Innocenzo, Enzo & Lucas, Andre, 2024. "Dynamic partial correlation models," Journal of Econometrics, Elsevier, vol. 241(2).
    20. Escribano, Alvaro & Sucarrat, Genaro, 2018. "Equation-by-equation estimation of multivariate periodic electricity price volatility," Energy Economics, Elsevier, vol. 74(C), pages 287-298.
    21. Simos G. Meintanis & John P. Nolan & Charl Pretorius, 2024. "Specification procedures for multivariate stable-Paretian laws for independent and for conditionally heteroskedastic data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 33(2), pages 517-539, June.
    22. Sucarrat, Genaro, 2018. "The Log-GARCH Model via ARMA Representations," MPRA Paper 100386, University Library of Munich, Germany.
    23. So, Mike K.P. & Chan, Thomas W.C. & Chu, Amanda M.Y., 2022. "Efficient estimation of high-dimensional dynamic covariance by risk factor mapping: Applications for financial risk management," Journal of Econometrics, Elsevier, vol. 227(1), pages 151-167.

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