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We consider a class of vector autoregressive models with banded coefficient matrices. This setting represents a type of sparse structure for high-dimensional time series, although the implied auto-covariance matrices are not banded. The structure is also practically meaningful when the component time series are ordered appropriately. We establish the convergence rates of the estimated banded autoregressive coefficient matrices. 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 structures for the auto-covariance functions of banded vector autoregressive processes, consistent estimators for the auto-covariance matrices are constructed.