  Differential Granger causality, that is understanding how Granger causal
relations differ between two related time series, is of interest in many
scientific applications. Modeling each time series by a vector autoregressive
(VAR) model, we propose a new method to directly learn the difference between
the corresponding transition matrices in high dimensions. Key to the new method
is an estimating equation constructed based on the Yule-Walker equation that
links the difference in transition matrices to the difference in the
corresponding precision matrices. In contrast to separately estimating each
transition matrix and then calculating the difference, the proposed direct
estimation method only requires sparsity of the difference of the two VAR
models, and hence allows hub nodes in each high-dimensional time series. The
direct estimator is shown to be consistent in estimation and support recovery
under mild assumptions. These results also lead to novel consistency results
with potentially faster convergence rates for estimating differences between
precision matrices of i.i.d observations under weaker assumptions than existing
results. We evaluate the finite sample performance of the proposed method using
simulation studies and an application to electroencephalogram (EEG) data.
