Margin-closed vector autoregressive time series models

Conditions are obtained for a Gaussian vector autoregressive time series of order $$, VAR($$), to have univariate margins that are autoregressive of order $$ or lower-dimensional margins that are also VAR($$). This can lead to $$-dimensional VAR($$) models that are closed with respect to a given partition $$ of $$ by specifying marginal serial dependence and some cross-sectional dependence parameters. The special closure property allows one to fit the subprocesses of multi-variate time series before assembling them by fitting the dependence structure between the subprocesses. We revisit the use of the Gaussian copula of the stationary joint distribution of observations in the VAR($$) 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.