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On moment conditions for quasi-maximum likelihood estimation of multivariate ARCH models

Author

Listed:
  • Marco Avarucci

    (Maastricht University)

  • Eric Beutner

    (Maastricht University)

  • Paolo Zaffaroni

    (Imperial College London and Università di Roma "La Sapienza")

Abstract

This paper questions whether it is possible to derive consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) for possibly the simplest VEC-GARCH model, namely the multivariate ARCH(1) model of the BEKK form, under weak moment conditions similar to the univariate case. In contrast to the univariate specification, we show that the expectation of the loglikelihood function is unbounded, away from the true parameter value, if (and only if) the observable has unbounded second moment. Despite this non-standard feature, consistency of the Gaussian QMLE is still warranted. The same moment condition proves to be necessary and sucient for the stationarity of the score, when evaluated at the true parameter value. This explains why high moment conditions, typically bounded sixth moment and above, have been used hitherto in the literature to establish the asymptotic normality of the QMLE in the multivariate framework.

Suggested Citation

  • Marco Avarucci & Eric Beutner & Paolo Zaffaroni, 2012. "On moment conditions for quasi-maximum likelihood estimation of multivariate ARCH models," DSS Empirical Economics and Econometrics Working Papers Series 2012/1, Centre for Empirical Economics and Econometrics, Department of Statistics, "Sapienza" University of Rome.
    • Handle: RePEc:sas:wpaper:20121
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. 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.
    3. Rasmus Pedersen & Olivier Wintenberger, 2017. "On the tail behavior of a class of multivariate conditionally heteroskedastic processes," Working Papers hal-01436267, HAL.
    4. Asai, M. & Chang, C-L. & McAleer, M.J. & Pauwels, L., 2018. "Asymptotic Theory for Rotated Multivariate GARCH Models," Econometric Institute Research Papers EI2018-38, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. Billio, Monica & Caporin, Massimiliano & Frattarolo, Lorenzo & Pelizzon, Loriana, 2023. "Networks in risk spillovers: A multivariate GARCH perspective," Econometrics and Statistics, Elsevier, vol. 28(C), pages 1-29.
    6. Nielsen, Heino Bohn & Rahbek, Anders, 2014. "Unit root vector autoregression with volatility induced stationarity," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 144-167.
    7. Christian Francq & Jean-Michel Zakoïan, 2016. "Estimating multivariate volatility models equation by equation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(3), pages 613-635, June.
    8. 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.
    9. Rasmus Søndergaard Pedersen & Olivier Wintenberger, 2017. "On the tail behavior of a class of multivariate conditionally heteroskedastic processes," Post-Print hal-01436267, HAL.
    10. Blasques, F. & Francq, Christian & Laurent, Sébastien, 2024. "Autoregressive conditional betas," Journal of Econometrics, Elsevier, vol. 238(2).
    11. Hetland, Simon & Pedersen, Rasmus Søndergaard & Rahbek, Anders, 2023. "Dynamic conditional eigenvalue GARCH," Journal of Econometrics, Elsevier, vol. 237(2).

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    Keywords

    multivariate ARCH models. moment conditions. VEC-GARCH.;

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