Granger causality from a time series 
to a time series 
, relative to a specified information set, is the property
that the past of 
improves prediction of 
beyond the information already contained in the past of
and any conditioning variables. More
formally, let 
be the sigma-algebra generated by the past of 
and conditioning variables 
, and let 
additionally contain the past of 
. Then 
does not Granger-cause 
if
 |
for every 
.
In a linear regression containing 
lags, or past values, of
each series,
 |
the corresponding null hypothesis is 
. Granger causality in both directions is
sometimes called feedback.
The result of a Granger-causality test depends on the selected lags. The lag order 
is the number of past time steps included. The result also depends on the sampling
interval, the conditioning variables, and the assumptions used to make the series
suitable for prediction. Granger causality is predictive rather than structural or
interventional causality: omitted variables, aggregation, or common drivers can produce
predictive precedence without a direct causal mechanism.
