Semiparametric Multivariate Volatility Models

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 Spéciaux de Recherche (FSR 05) of the Université catholique de Louvain. The authors thank three anonymous referees for valuable comments and suggestions and Luc Bauwens, Geert Dhaene, Feico Drost, Wolfgang Härdle, 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.