Granger causality in the frequency domain: derivation and applications

TL;DR

This paper introduces a statistical approach to measure causality between time series, specifically Granger causality, and derives its calculation methods in both time and frequency domains, with applications across various fields.

Contribution

It provides a comprehensive derivation of Granger causality in the frequency domain and illustrates its calculation with non-parametric estimation methods, expanding its applicability.

Findings

Derivation of Granger causality in the frequency domain

Numerical examples demonstrating non-parametric estimation

Discussion of limitations and alternative causality measures

Abstract

Physicists are starting to work in areas where noisy signal analysis is required. In these fields, such as Economics, Neuroscience, and Physics, the notion of causality should be interpreted as a statistical measure. We introduce to the lay reader the Granger causality between two time series and illustrate ways of calculating it: a signal $X$ ``Granger-causes'' a signal $Y$ if the observation of the past of $X$ increases the predictability of the future of $Y$ when compared to the same prediction done with the past of $Y$ alone. In other words, for Granger causality between two quantities it suffices that information extracted from the past of one of them improves the forecast of the future of the other, even in the absence of any physical mechanism of interaction. We present derivations of the Granger causality measure in the time and frequency domains and give numerical examples using a non-parametric estimation method in the frequency domain. Parametric methods are addressed in the Appendix. We discuss the limitations and applications of this method and other alternatives to measure causality.