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Abstract: In literature on multivariate modeling using copulas, it is typical to first model univariate margins and then the multivariate dependence structure between the marginal components. This is useful because there are many diagnostics that can help in selecting univariate models. Following the same idea, we derive the conditions for when a multivariate stationary Gaussian vector autoregressive (VAR) time series is closed under margins, i.e., it has univariate autoregressive (AR) margins or lower-dimensional vector autoregressive (VAR) margins. It leads to a copula model and it can be extended to a regime-switching setting. The constraint of the closure under margins can reduce the number of parameters in VAR and Markov switching vector autoregressive (MSVAR) models. Moreover, after transforming the stationary univariate margins into standard Gaussian, the property of closure under margins enables a new framework of modeling high-dimensional time series by modeling its low-dimensional sub-processes first and then modeling their dependence structure. The framework makes it more flexible in analyzing marginal behavior of the multivariate time series and also enables a multi-stage estimation procedure.