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Semiparametric Multivariate Volatility Models

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  • Hafner, Christian M.
  • Rombouts, Jeroen V.K.

Abstract

We consider a model for a multivariate time series where the conditional covariance matrix is a function of a finite-dimensional parameter and the innovation distribution is nonparametric. The semiparametric lower bound for the estimation of the euclidean parameter is characterized, and it is shown that adaptive estimation without reparametrization is not possible. Based on a consistent first-stage estimator (such as quasi maximum likelihood), we propose a semiparametric estimator that estimates the efficient influence function using kernel estimators. We state conditions under which the estimator attains the semiparametric lower bound. For particular models such as the constant conditional correlation model, adaptive estimation of the dynamic part of the model is shown to be possible. To avoid the curse of dimensionality one can, e.g., restrict the multivariate density to the class of spherical distributions, for which we also derive the semiparametric efficiency bound and an estimator that attains this bound. A simulation experiment demonstrates the efficiency gain of the proposed estimator compared with quasi maximum likelihood estimation.Rombouts work was supported by the Centre for Research on e-Finance, HEC Montreal. Hafner gratefully acknowledges financial support by the Fonds Sp ciaux de Recherche (FSR 05) of the Universit catholique de Louvain. The authors thank three anonymous referees for valuable comments and suggestions and Luc Bauwens, Geert Dhaene, Feico Drost, Wolfgang H rdle, Douglas Hodgson, Jens Peter Kreiss, Oliver Linton, and Bas Werker for helpful discussions. We also thank participants of the CORE econometrics seminar, the York annual meeting in econometrics, the annual econometric study group meeting 2002 in Bristol, the 2003 workshop The Art of Semiparametrics in Berlin, and the statistics seminar of the Stockholm School of Economics for valuable comments.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 23 (2007)
Issue (Month): 02 (April)
Pages: 251-280

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Handle: RePEc:cup:etheor:v:23:y:2007:i:02:p:251-280_07

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  1. Newey, Whitney K, 1990. "Semiparametric Efficiency Bounds," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 5(2), pages 99-135, April-Jun.
  2. Oliver Linton & Douglas J.Hodgson & Keith Vorkink, 2001. "Testing the Capital Asset Pricing Model Efficiently Under Elliptical Symmetry: A Semiparametric Approach," FMG Discussion Papers dp382, Financial Markets Group.
  3. HAFNER, Christian M. & HERWARTZ, Helmut, 1998. "Volatility impulse response functions for multivariate GARCH models," CORE Discussion Papers 1998047, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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  7. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer, vol. 52(3), pages 471-480, September.
  8. BAUWENS, Luc & LAURENT, Sébastien & ROMBOUTS, Jeroen, 2003. "Multivariate GARCH models: a survey," CORE Discussion Papers 2003031, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  9. Gonzalez-Rivera, Gloria & Drost, Feike C., 1999. "Efficiency comparisons of maximum-likelihood-based estimators in GARCH models," Journal of Econometrics, Elsevier, vol. 93(1), pages 93-111, November.
  10. Drost, F.C. & Klaasens, C.A.J. & Werker, B.J.M., 1994. "Adaptive Estimation in Time Series Models," Papers 9488, Tilburg - Center for Economic Research.
  11. Gloria Gonzalez-Rivera, 1997. "A note on adaptation in garch models," Econometric Reviews, Taylor & Francis Journals, vol. 16(1), pages 55-68.
  12. Drost, F.C. & Klaassen, C.A.J., 1996. "Efficient Estimation in Semiparametric GARCH Models," Discussion Paper 1996-38, Tilburg University, Center for Economic Research.
  13. Steigerwald, Douglas G., 1992. "Adaptive estimation in time series regression models," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 251-275.
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  15. Oliver Linton, 1993. "Adaptive Estimation in ARCH Models," Cowles Foundation Discussion Papers 1054, Cowles Foundation for Research in Economics, Yale University.
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  17. Engle, Robert F & Gonzalez-Rivera, Gloria, 1991. "Semiparametric ARCH Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(4), pages 345-59, October.
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Cited by:
  1. Annastiina Silvennoinen & Timo Teräsvirta, 2008. "Multivariate GARCH models," CREATES Research Papers 2008-06, School of Economics and Management, University of Aarhus.
  2. Jeroen V.K. Rombouts & Marno Verbeek, 2004. "Evaluating Portfolio Value-at-Risk using Semi-Parametric GARCH Models," Cahiers de recherche 04-14, HEC Montréal, Institut d'économie appliquée.
  3. Gabriele Fiorentini & Enrique Sentana, 2007. "On the efficiency and consistency of likelihood estimation in multivariate conditionally heteroskedastic dynamic regression models," Working Paper Series 38-07, The Rimini Centre for Economic Analysis, revised Jul 2007.
  4. Hafner, Christian M. & Preminger, Arie, 2009. "Asymptotic Theory For A Factor Garch Model," Econometric Theory, Cambridge University Press, vol. 25(02), pages 336-363, April.
  5. Jensen, Mark J. & Maheu, John M., 2013. "Bayesian semiparametric multivariate GARCH modeling," Journal of Econometrics, Elsevier, vol. 176(1), pages 3-17.
  6. Hafner, Christian M. & Preminger, Arie, 2009. "On asymptotic theory for multivariate GARCH models," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2044-2054, October.
  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. Gabriele Fiorentini & Enrique Sentana, 2010. "Dynamic Specification Tests for Static Factor Models," Working Paper Series 04_10, The Rimini Centre for Economic Analysis.
  9. Gabriele Fiorentini & Enrique Sentana, 2012. "Tests For Serial Dependence In Static, Non-Gaussian Factor Models," Working Papers wp2012_1211, CEMFI.

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