A novel estimation approach for a general class of semi-parametric multivariate time series models is introduced where the conditional mean is modeled through parametric functions. The focus of the estimation is the conditional mean parameter vector for non-negative time series. 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. Novel two-stage Multivariate Weighted Least Square Estimators (MWLSEs) are introduced which enjoy the same consistency property as the QMLEs but provide improved efficiency with a suitable choice of the weighting sequence of matrices in the second stage. The proposed method enables a more accurate estimation of model parameters, particularly for data where maximum likelihood estimation is infeasible. Moreover, consistency and asymptotic normality of MWLSEs are derived, and their efficiency is proved under the correct specification of the weighting sequence. The estimation performance of QMLE and MWLSE is also compared through simulation experiments and a real data application, showing the superior accuracy of the proposed methodology.