  As a special infinite-order vector autoregressive (VAR) model, the vector
autoregressive moving average (VARMA) model can capture much richer temporal
patterns than the widely used finite-order VAR model. However, its practicality
has long been hindered by its non-identifiability, computational
intractability, and difficulty of interpretation, especially for
high-dimensional time series. This paper proposes a novel sparse infinite-order
VAR model for high-dimensional time series, which avoids all above drawbacks
while inheriting essential temporal patterns of the VARMA model. As another
attractive feature, the temporal and cross-sectional structures of the
VARMA-type dynamics captured by this model can be interpreted separately, since
they are characterized by different sets of parameters. This separation
naturally motivates the sparsity assumption on the parameters determining the
cross-sectional dependence. As a result, greater statistical efficiency and
interpretability can be achieved with little loss of temporal information. We
introduce two $\ell_1$-regularized estimation methods for the proposed model,
which can be efficiently implemented via block coordinate descent algorithms,
and derive the corresponding nonasymptotic error bounds. A consistent model
order selection method based on the Bayesian information criteria is also
developed. The merit of the proposed approach is supported by simulation
studies and a real-world macroeconomic data analysis.
