Mathematical statement

Let y and x be anchored time series. To analysis the absent antecedent that x does not Granger-cause y, one aboriginal finds the able lagged ethics of y to accommodate in a univariate autoregression of y:

yt = a0 + a1yt − 1 + a2yt − 2 + ... + amyt − m + residualt.

Here yt − j is retained in the corruption if and alone if it has a cogent t-statistic; m is the greatest lag breadth for which the lagged abased capricious is significant.

Next, the autoregression is aggrandized by including lagged ethics of x:

yt = a0 + a1yt − 1 + a2yt − 2 + ...amyt − m + bpxt − p + ... + bqxt − q + residualt.

One retains in this corruption all lagged ethics of x that are alone cogent according to their t-statistics, provided that collectively they add allegorical ability to the corruption according to an F-test (whose absent antecedent is no allegorical ability accordingly added by the x's). In the characters of the aloft aggrandized regression, p is the shortest, and q is the longest, lag breadth for which the lagged amount of x is significant.

The absent antecedent that x does not Granger-cause y is accustomed if and alone if no lagged ethics of x are retained in the regression.