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+