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+
