Conditional Granger causality and partitioned Granger causality: differences and similarities

Neural information modeling and analysis often requires a measurement of the mutual influence among many signals. A common technique is the conditional Granger causality (cGC) which measures the influence of one time series on another time series in the presence of a third. Geweke has translated this condition into the frequency domain and has explored the mathematical relationships between the time and frequency domain expressions. Chen has observed that in practice, the expressions may return (meaningless) negative numbers, and has proposed an alternative which is based on a partitioned matrix scheme, which we call partitioned Granger causality (pGC). There has been some confusion in the literature about the relationship between cGC and pGC; some authors treat them as essentially identical measures, while others have noted that some properties (such as the relationship between the time and frequency domain expressions) do not hold for the pGC. This paper presents a series of matrix equalities that simplify the calculation of the pGC. In this simplified expression, the essential differences and similarities between the cGC and the pGC become clear; in essence, the pGC is dependent on only a subset of the parameters in the model estimation, and the noise residuals (which are uncorrelated in the cGC) need not be uncorrelated in the pGC. The mathematical results are illustrated with a simulation, and the measures are applied to an EEG dataset.