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...

This paper demonstrates that all rank test statistics are functions of implicit null space estimators. The paper proposes a novel theory of null space estimation that allows for standard asymptotics, polynomial regressions, and cointegration asymptotics. The paper proves that the behaviour of...