Granger Causality

Granger causality formalizes a predictive notion of directed influence: a signal X Granger-causes a signal Y if the past of X significantly improves the prediction of Y beyond what Y's own past already provides. It is typically implemented by fitting a multivariate autoregressive model, comparing the residual prediction-error variance of the restricted model (Y from its own past) against the full model (Y from its past and X's), and testing the reduction, often after a spectral decomposition (Geweke) to obtain frequency-resolved influence.

Its assumptions are strict and consequential: covariance-stationarity over the analysis window, a well-chosen model order, adequate whitening, and no important unmodeled common driver. Violations are the norm in neural data, and it is well documented that additive measurement noise, low sampling rate, and filtering can bias Granger estimates and even invert the apparent direction. Multivariate (conditional) Granger causality partially controls for a third region's influence but does not rescue the method from unobserved common inputs.