This source includes a helpful summary of Granger causality, written by Granger himself. http://www.scholarpedia.org/article/Granger_causality

The topic of how to define causality has kept philosophers busy for over two thousand years and has yet to be resolved. It is a deep convoluted question with many possible answers which do not satisfy everyone, and yet it remains of some importance. Investigators would like to think that they have found a "cause", which is a deep fundamental relationship and possibly potentially useful.

In the early 1960's I was considering a pair of related stochastic processes which were clearly inter-related and I wanted to know if this relationship could be broken down into a pair of one way relationships. It was suggested to me to look at a definition of causality proposed by a very famous mathematician, Norbert Weiner, so I adapted this definition (Wiener 1956) into a practical form and discussed it.

Applied economists found the definition understandable and useable and applications of it started to appear. However, several writers stated that "of course, this is not real causality, it is only Granger causality." Thus, from the beginning, applications used this term to distinguish it from other possible definitions.

The basic "Granger Causality" definition is quite simple. Suppose that we have three terms, Xt , Yt , and Wt , and that we first attempt to forecast Xt+1 using past terms of Xt and Wt . We then try to forecast Xt+1 using past terms of Xt , Yt , and Wt . If the second forecast is found to be more successful, according to standard cost functions, then the past of Y appears to contain information helping in forecasting Xt+1 that is not in past Xt or Wt . In particular, Wt could be a vector of possible explanatory variables. Thus, Yt would "Granger cause" Xt+1 if (a) Yt occurs before Xt+1 ; and (b) it contains information useful in forecasting Xt+1 that is not found in a group of other appropriate variables.

Naturally, the larger Wt is, and the more carefully its contents are selected, the more stringent a criterion Yt is passing. Eventually, Yt might seem to contain unique information about Xt+1 that is not found in other variables which is why the "causality" label is perhaps appropriate.

The definition leans heavily on the idea that the cause occurs before the effect, which is the basis of most, but not all, causality definitions. Some implications are that it is possible for Yt to cause Xt+1 and for Xt to cause Yt+1 , a feedback stochastic system. However, it is not possible for a determinate process, such as an exponential trend, to be a cause or to be caused by another variable.

It is possible to formulate statistical tests for which I now designate as G-causality, and many are available and are described in some econometric textbooks (see also the following section and the #references). The definition has been widely cited and applied because it is pragmatic, easy to understand, and to apply. It is generally agreed that it does not capture all aspects of causality, but enough to be worth considering in an empirical test.

There are now a number of alternative definitions in economics, but they are little used as they are less easy to implement.