Mathematical Statement
Let y and x be stationary time series. To test the null hypothesis that x does not Granger-cause y, one first finds the proper lagged values of y to include in a univariate autoregression of y:
Next, the autoregression is augmented by including lagged values of x:
One retains in this regression all lagged values of x that are individually significant according to their t-statistics, provided that collectively they add explanatory power to the regression according to an F-test (whose null hypothesis is no explanatory power jointly added by the x's). In the notation of the above augmented regression, p is the shortest, and q is the longest, lag length for which the lagged value of x is significant.
The null hypothesis that x does not Granger-cause y is accepted if and only if no lagged values of x are retained in the regression.
Read more about this topic: Granger Causality
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