State Space Methods for Granger-Geweke Causality Measures

At least two recent developments have put the spotlight on some significant gaps in the theory of multivariate time series. The recent interest in the dynamics of networks; and the advent, across a range of applications, of measuring modalities that operate on different temporal scales. Fundamental to the description of network dynamics is the direction of interaction between nodes, accompanied by a measure of the strength of such interactions. Granger causality (GC) and its associated frequency domain strength measures (GEMs) (due to Geweke) provide a framework for the formulation and analysis of these issues. In pursuing this setup, three significant unresolved issues emerge. Firstly computing GEMs involves computing submodels of vector time series mod- els, for which reliable methods do not exist; Secondly the impact of filtering on GEMs has never been definitively established. Thirdly the impact of downsampling on GEMs has never been established. In this work, using state space methods, we resolve all these issues and illustrate the results with some simulations. Our discussion is motivated by some problems in (fMRI) brain imaging but is of general applicability.