Margin-closed vector autoregressive time series models

Conditions are obtained for a Gaussian vector autoregressive time series of order $k$, VAR($k$), to have univariate margins that are autoregressive of order $k$ or lower-dimensional margins that are also VAR($k$). This can lead to $d$-dimensional VAR($k$) models that are closed with respect to a given partition $\{S_1,\ldots,S_n\}$ of $\{1,\ldots,d\}$ by specifying marginal serial dependence and some cross-sectional dependence parameters. The special closure property allows one to fit the sub-processes of multivariate time series before assembling them by fitting the dependence structure between the sub-processes. We revisit the use of the Gaussian copula of the stationary joint distribution of observations in the VAR($k$) process with non-Gaussian univariate margins but under the constraint of closure under margins. This construction allows more flexibility in handling higher-dimensional time series and a multi-stage estimation procedure can be used. The proposed class of models is applied to a macro-economic data set and compared with the relevant benchmark models.