# Granger causality in the frequency domain: derivation and applications

arXiv: https://arxiv.org/abs/2106.03990 · HTML: https://mlsanity.com/paper/2106.03990/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2021.

## Built on by (most cited first, 2 of 2)

- [Causal Representation Meets Stochastic Modeling under Generic Geometry](https://mlsanity.com/paper/2602.05033/) (2026)
- [Beyond Performance Scores: Directed Functional Connectivity as a Brain-Based Biomarker for Motor Skill Learning and Retention](https://mlsanity.com/paper/2502.14731/) (2025)

## Abstract (arXiv)

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.