Relationship Between Granger Noncausality and Network Graph of State-Space Representations

The goal of this paper is to explore the relationship between the network graph of a state-space representation of an observed process and the causal relations among the components of that process. We will show that the existence of a linear time-invariant state-space representation, with its network graph being the star graph, is equivalent to (conditional) Granger noncausal relations among the components of the output process. Granger noncausality is a statistical concept, which applies to arbitrary processes and does not depend on the representation of the process. That is, we relate intrinsic properties of a process with the network graph of its state-space representations.