This paper is devoted to a model for analysis and forecasting of vector time series and the corresponding procedure of Bayesian sequential estimation. This model can also be viewed as a multivariate extension of the (univariate) seasonal growth multiplicative model(Harrison, 1965; Migon, 1984). The basic structure of this multivariate model consists of a locally linear trend component for each individual series and a shared multiplicative seasonal component, common to all marginal series.The procedure of sequential estimation is based on analytic approximations to obtain a conjugate analysis and represents a nonlinear extension of the algorithm presented by Barbosa and Harrison (1992). Details of the proposed procedure and its practical implementation are shown, and two numerical examples are provided.