Behaviour of Granger causality under filtering: Theoretical invariance and practical application

Granger causality (G-causality) is increasingly employed as a method for identifying directed functional connectivity in neural time series data. However, little attention has been paid to the influence of common preprocessing methods such as filtering on G-causality inference. Filtering is often used to remove artifacts from data and/or to isolate frequency bands of interest. Here, we show [following Geweke (1982)] that G-causality for a stationary vector autoregressive (VAR) process is fully invariant under the application of an arbitrary invertible filter; therefore filtering cannot and does not isolate frequency-specific G-causal inferences. We describe and illustrate a simple alternative: integration of frequency domain (spectral) G-causality over the appropriate frequencies (“band limited G-causality”). We then show, using an analytically solvable minimal model, that in practice G-causality inferences often do change after filtering, as a consequence of large increases in empirical model order induced by filtering. Finally, we demonstrate a valid application of filtering in removing a nonstationary (“line noise”) component from data. In summary, when applied carefully, filtering can be a useful preprocessing step for removing artifacts and for furnishing or improving stationarity; however filtering is inappropriate for isolating causal influences within specific frequency bands.

► Granger causality is theoretically invariant under an arbitrary multivariate, stable, invertible filter. ► Therefore, filtering cannot isolate frequency-specific Granger-causal effects. ► To isolate frequency-specific causal effects, as an alternative to filtering we introduce “band-limited” Granger causality. ► Estimated in sample from time series, filtering degrades causal inference, due (mainly) to an increase in empirical AR model order. ► The filter invariance applies strictly to stationary time series; filtering remains a valid tool to improve stationarity (e.g., to remove line noise).