Representing conditional Granger causality by vector auto-regressive parameters

Granger Causality (GC) has been widely applied to various scientific fields to reveal causal relationships between dynamical variables. The mathematical framework of GC is based on the vector auto-regression (VAR) model, and the GC value from one variable to the other is defined as the logarithmic ratio of the variance of two prediction errors obtained by excluding and including the second variable in the VAR model respectively. Besides its definition, GC shall also be reflected in the regression parameters of the VAR model, e.g., larger regression coefficients indicate stronger causal interactions in general. Yet an explicit description of how the GC value depends on the VAR parameters for a general multi-variable case remains lacking. In this work, we aim to bridge this gap by expressing conditional GC using the VAR parameters, which provides an alternative interpretation of GC with novel intuition. The analysis developed in this work may also benefit the study of the VAR model in the future.