A Measure of Prediction Precision for Granger Causality Analysis

Identification of causal interactions from time-series data is an emerging issue in many fields of science. The causal connection between two variables is detected by the Granger causality if the prediction of one variable based on a linear combination of its past values can be improved by incorporating past values of another. The mean squared deviation is usually used to measure prediction for the Granger causality analysis. In this contribution, it is shown on an example that the mean squared deviation strictly decreases by adding a past value even from a causally independent variable and, due to this behaviour, it is unusable for such causal analysis. On the other hand, the adjusted squared deviation does not collide with the Granger causality definition in this way. The presented results imply that the confidence of the commonly used F-test of prediction improvement in the Granger causality analysis depends on the number of past values of the predicted variable included in a linear prediction. The adjusted squared deviation can be used to determine the proper number of past values.