In multivariate time series analysis, understanding the underlying causal relationships among variables is often of interest for various applications. Directed acyclic graphs (DAGs) provide a powerful framework for representing causal dependencies. This paper proposes a novel Bayesian approach for modeling multivariate time series where conditional independencies and causal structure are encoded by a DAG. The proposed model allows structural properties such as stationarity to be easily accommodated. Given the application, we further extend the model for matrix-variate time series. We take a Bayesian approach to inference, and a ``immersion-posterior'' based efficient computational algorithm is developed. The posterior convergence properties of the proposed method are established along with two identifiability results for the unrestricted structural equation models. The utility of the proposed method is demonstrated through simulation studies and real data analysis.

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