Granger Causality

Granger causality allows to test if a variable X is causally linked to another variable Y. We say Y G-causes X if predicting X with only past information of X gives worse results then using past values of X and Y. Clearly X and Y have to be defined over time. Which means X and Y are time series. If X is causally linked with Y, then we can can model X as the following linear Auto Regressive model:

The prediction is the sum of the AR model for X, the AR model for Y and an error term. The idea is if the error for the prediction is minimal using the above predictor, then Y G-causes X if the weight for y is significantly different from 0. As one can see if the weight is 0 then Y has no effect on the predictor at all, so
it is not causing X. So if we want to test if Y G-causes X given some data, we estimate the weights and perform a F-Test of the weight vector of Y and 0. So our null-hypothesis is:

In conclusion we can test if two variables X and Y are causally linked by performing a statistical test on the weights (or influence) of one variable to another.