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.

翻译：本文给出了$k$阶高斯向量自回归时间序列VAR($k$)具备$k$阶自回归单变量边际或同样为VAR($k$)的低维边际所满足的条件。这可以构建对给定划分$\{S_1,\ldots,S_n\}$（$\{1,\ldots,d\}$的划分）封闭的$d$维VAR($k$)模型，通过指定边际序列依赖性和部分横截面依赖参数实现。这种特殊的封闭性允许先拟合多元时间序列的子过程，再通过拟合子过程间的依赖结构进行整合。我们重新审视了在非高斯单变量边际但满足边际封闭约束条件下，VAR($k$)过程观测值平稳联合分布的高斯Copula的应用。该构造在处理高维时间序列方面具有更强的灵活性，并可采用多阶段估计方法。我们将所提出的模型类别应用于宏观经济数据集，并与相关基准模型进行了比较。