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The article proposes a Bayesian methodology to implement complete and cohesive analysis of vector time series. Assuming the series are generated by vector autoregressive moving average process, the identification, diagnostic checking, estimation and forecasting phases of time series analysis are done by referring to the appropriate posterior or predictive distributions. The identification phase is based on approximating the posterior distribution of the coefficients of the largest possible model by a matric-variate generalization of the t-distribution (matrix t-distribution). Then the insignificant coefficients are eliminated by a sequence of F or Chi-square tests using a procedure similar to the backward elimination technique used in regression analysis. The diagnostic checking phase is done using overfitting tests, the estimation phase is done using the matrix t and Wishart distributions, and the forecasting phase is performed using multivariate t-distribution. Using the proposed posterior distributions, the article proposes the machinery necessary to implement the four phases of multivariate time series analysis. The proposed Bayesian methodology has been illustrated and checked by a simulated data used by distinguished researchers. The Initial examination of the numerical results supports the adequacy of using the proposed methodology. [ABSTRACT FROM AUTHOR]