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.

翻译：差异引力因果性,即理解Granger因果关系在两个相关时间序列之间有何差异,是许多科学应用中感兴趣的。用矢量自动递减模型(VAR)模型来模拟每个时间序列,我们提出一种新方法,直接了解高维相应过渡矩阵之间的差异。新方法的关键是,根据Yule-Walker方程式构建的估算方程式,该方程式将过渡矩阵的差异与相应精确矩阵的差异联系起来。与分别估计每个过渡矩阵和随后计算差异相比,拟议的直接估算方法只要求两个VAR模型的差异宽度,从而允许在每个高维时间序列中建立中心节点。直接估计数字显示,在轻度假设下估算和支持恢复方面是一致的。这些结果还导致新的一致性结果,在估计i.d观察在比现有结果更弱的情况下的精确矩阵之间的差异时,有可能更快的趋同率。我们利用模拟研究和电脑图数据应用来评估拟议方法的有限样本性表现。