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

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.