New considerations on the validity of the Wiener-Granger causality test

Abstract

The Wiener-Granger causality test is used to predict future experimental results from past observations in a purely mathematical way. For instance, in many scientific papers this test has been used to study the causality relations in the case of neuronal activities. Albeit some papers reported repeatedly about problems or open questions related to the application of the Granger causality test on biological systems, these criticisms were always related to some kind of assumptions to be made before the test's application. In our paper instead we investigate the Granger method itself, making use exclusively of fundamental mathematical tools like Fourier transformation and differential calculus. We find that the ARMA method reconstructs any time series from any time series, regardless of their properties, and that the quality of the reconstruction is given by the properties of the Fourier transform. In literature several definitions of “causality” have been proposed in order to maintain the idea that the Granger test might be able to predict future events and prove causality between time series. We find instead that not even the most fundamental requirement underlying any possible definition of causality is met by the Granger causality test. No matter of the details, any definition of causality should refer to the prediction of the future from the past; instead by inverting the time series we find that Granger also allows one to “predict” the past from the future.

Letter to the editor of Heliyon re: new considerations on the validity of the Wiener-Granger causality test [Heliyon 6(10) (2020) e05208]

“…Dr. Martina Chvosteková and her collaborator Dr. Anna Krakovská have criticized the conclusion of the recently published article ( Grassmann, 2020 ), which states: " ...not even the most basic requirement underlying any possible definition of causality is met by the Granger causality test... ". I am very glad about their contribution, since the topic is important and will profit from as much discussion as possible.…”

“…To be as clear as possible and show how these choices are of no relevance for the conclusion, the main points of ( Grassmann, 2020 ) (which were not questioned by Dr. M. Chvosteková and Dr. A. Krakovská) will be presented in the following, without making use of the RMS of the neural signals at all:…”

Letter to the editor of Heliyon re: Grassmann, G. “New considerations on the validity of the Wiener-Granger causality test” [Heliyon 6 (2020) e05208]

“…Dear Editor, a recently published article [1] questions the Wiener-Granger causality test, more commonly referred to as the Granger causality test. Although the abstract of the paper states that the results were obtained by mathematical tools such as the Fourier transform and differential calculus, the presented serious claim that "...not even the most basic requirement underlying any possible definition of causality is met by the Granger causality test..." is in fact based on a questionable numerical testing of two filtered neural signals (A and B).…”

“…It is said that a time series 𝑥 =∶ {𝑥 𝑡 } 𝑇 𝑡=1 Granger causes a time series 𝑦 =∶ {𝑦 𝑡 } 𝑇 𝑡=1 if the variance (or, equivalently the standard deviation) of the prediction error of 𝑦 including only past values of 𝑦 𝜖 𝑡 (𝑦∕𝑦) = 𝑦 𝑡 − 𝑝 ∑ 𝑗=1 𝛼 𝑗 𝑦 𝑡−𝑗 (1) is greater than the variance of the prediction error of 𝑦 including jointly the past values of both 𝑦 and 𝑥…”

“…In [1], instead of the 𝐹 -test (or any statistical test), the causal link between time series is inferred after just comparing the standard deviation estimates of the predictive errors, i.e. 𝜎(𝜖 𝑡 (𝑦∕𝑦)) and 𝜎(𝜖 𝑡 (𝑦∕𝑦, 𝑥)) to each other.…”