*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.

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