  In this work a general semi-parametric multivariate model where the first two
conditional moments are assumed to be multivariate time series is introduced.
The focus of the estimation is the conditional mean parameter vector for
discrete-valued distributions. Quasi-Maximum Likelihood Estimators (QMLEs)
based on the linear exponential family are typically employed for such
estimation problems when the true multivariate conditional probability
distribution is unknown or too complex. Although QMLEs provide consistent
estimates they may be inefficient. In this paper novel two-stage Multivariate
Weighted Least Square Estimators (MWLSEs) are introduced which enjoy the same
consistency property as the QMLEs but can provide improved efficiency with
suitable choice of the covariance matrix of the observations. The proposed
method allows for a more accurate estimation of model parameters in particular
for count and categorical data when maximum likelihood estimation is
unfeasible. Moreover, consistency and asymptotic normality of MWLSEs are
derived. The estimation performance of QMLEs and MWLSEs is compared through
simulation experiments and a real data application, showing superior accuracy
of the proposed methodology.
