Geometric and long run aspects of Granger causality
将多元格兰杰因果关系扩展到考虑因果发生的子空间和长期格兰杰因果关系，推导了这些新概念的性质，并证明均值回归、协整、可控性和线性理性预期都是其特例。
This paper extends multivariate Granger causality to take into account the subspaces along which Granger causality occurs as well as long run Granger causality. The properties of these new notions of Granger causality, along with the requisite restrictions, are derived and extensively studied for a wide variety of time series processes including linear invertible processes and VARMA. Using the proposed extensions, the paper demonstrates that: (i) mean reversion in is an instance of long run Granger non-causality, (ii) cointegration is a special case of long run Granger non-causality along a subspace, (iii) controllability is a special case of Granger causality, and finally (iv) linear rational expectations entail (possibly testable) Granger causality restriction along subspaces.