  High-dimensional multivariate time series are challenging due to the
dependent and high-dimensional nature of the data, but in many applications
there is additional structure that can be exploited to reduce computing time
along with statistical error. We consider high-dimensional vector
autoregressive processes with spatial structure, a simple and common form of
additional structure. We propose novel high-dimensional methods that take
advantage of such structure without making model assumptions about how distance
affects dependence. We provide non-asymptotic bounds on the statistical error
of parameter estimators in high-dimensional settings and show that the proposed
approach reduces the statistical error. An application to air pollution in the
US demonstrates that the estimation approach reduces both computing time and
prediction error and gives rise to results that are meaningful from a
scientific point of view, in contrast to high-dimensional methods that ignore
spatial structure. In practice, these high-dimensional methods can be used to
decompose high-dimensional multivariate time series into lower-dimensional
multivariate time series that can be studied by other methods in more depth.
