Granger causality graphs for multivariate time series

In this paper, we discuss the properties of mixed graphs whichvisualize causal relationships between the components of multivariatetime series. In these Granger-causality graphs, the vertices, representing thecomponents of the time series, are connected by arrows according to theGranger-causality relations between the variables whereas lines correspondto contemporaneous conditional association. We show that the concept ofGranger-causality graphs provides a framework for the derivation ofgeneral noncausality relations relative to reduced information sets by performingsequences of simple operations on the graphs. We briefly discussthe implications for the identification of causal relationships. Finally we provide an extension of the linear concept to strong Granger-causality