  We consider a class of vector autoregressive models with banded coefficient
matrices. The setting represents a type of sparse structure for
high-dimensional time series, though the implied autocovariance matrices are
not banded. The structure is also practically meaningful when the order of
component time series is arranged appropriately. The convergence rates for the
estimated banded autoregressive coefficient matrices are established. We also
propose a Bayesian information criterion for determining the width of the bands
in the coefficient matrices, which is proved to be consistent. By exploring
some approximate banded structure for the auto-covariance functions of banded
vector autoregressive processes, consistent estimators for the auto-covariance
matrices are constructed.
