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

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Author Info

  • 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.

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File URL: http://www.dss.uniroma1.it/RePec/sas/wpaper/20121_ABZ.pdf
File Function: First version, 2012
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Bibliographic Info

Paper provided by Centre for Empirical Economics and Econometrics, Department of Statistics, "Sapienza" University of Rome in its series DSS Empirical Economics and Econometrics Working Papers Series with number 2012/1.

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Length: 21 pages
Date of creation: Jan 2012
Date of revision:
Handle: RePEc:sas:wpaper:20121

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

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References

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  1. Abadir,Karim M. & Magnus,Jan R., 2005. "Matrix Algebra," Cambridge Books, Cambridge University Press, number 9780521537469, December.
  2. repec:cup:cbooks:9780521496032 is not listed on IDEAS
  3. Jensen, S ren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(06), pages 1203-1226, December.
  4. 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.
  5. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
  6. Nelson, Daniel B., 1990. "Stationarity and Persistence in the GARCH(1,1) Model," Econometric Theory, Cambridge University Press, vol. 6(03), pages 318-334, September.
  7. Annastiina Silvennoinen & Timo Teräsvirta, 2008. "Multivariate GARCH models," CREATES Research Papers 2008-06, School of Economics and Management, University of Aarhus.
  8. Stelzer, Robert, 2008. "On The Relation Between The Vec And Bekk Multivariate Garch Models," Econometric Theory, Cambridge University Press, vol. 24(04), pages 1131-1136, August.
  9. HAFNER, Christian M. & PREMINGER, Arie, 2006. "Asymptotic theory for a factor GARCH model," CORE Discussion Papers 2006071, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  10. Roberts, Leigh A., 1995. "On the Existence of Moments of Ratios of Quadratic Forms," Econometric Theory, Cambridge University Press, vol. 11(04), pages 750-774, August.
  11. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(01), pages 29-52, March.
  12. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(01), pages 122-150, February.
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Cited by:
  1. Rasmus Søndergaard Pedersen & Anders Rahbek, 2012. "Multivariate Variance Targeting in the BEKK-GARCH Model," CREATES Research Papers 2012-53, School of Economics and Management, University of Aarhus.
  2. Anders Rahbek & Heino Bohn Nielsen, 2012. "Unit root vector autoregression with volatility induced stationarity," Discussion Papers 12-02, University of Copenhagen. Department of Economics.

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