  Vector autoregression has been widely used for modeling and analysis of
multivariate time series data. In high-dimensional settings, model parameter
regularization schemes inducing sparsity yield interpretable models and
achieved good forecasting performance. However, in many data applications, such
as those in neuroscience, the Granger causality graph estimates from existing
vector autoregression methods tend to be quite dense and difficult to
interpret, unless one compromises on the goodness-of-fit. To address this
issue, this paper proposes to incorporate a commonly used structural assumption
-- that the ground-truth graph should be largely connected, in the sense that
it should only contain at most a few components. We take a Bayesian approach
and develop a novel tree-rank prior distribution for the regression
coefficients. Specifically, this prior distribution forces the non-zero
coefficients to appear only on the union of a few spanning trees. Since each
spanning tree connects $p$ nodes with only $(p-1)$ edges, it effectively
achieves both high connectivity and high sparsity. We develop a computationally
efficient Gibbs sampler that is scalable to large sample size and high
dimension. In analyzing test-retest functional magnetic resonance imaging data,
our model produces a much more interpretable graph estimate, compared to
popular existing approaches. In addition, we show appealing properties of this
new method, such as efficient computation, mild stability conditions and
posterior consistency.
