# Granger Causality

The Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another. A time series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included), that those X values provide statistically significant information about future values of Y.

The test for Granger causality works by first doing a regression of ΔY on lagged values of ΔY. (Here ΔY is the first difference of the variable Y — that is, Y minus its one-period-prior value. The regressions are performed in terms of ΔY rather than Y if Y is not stationary but ΔY is.) Once the set of significant lagged values for ΔY is found (via t-statistics or p-values), the regression is augmented with lagged levels of ΔX. Any particular lagged value of ΔX is retained in the regression if (1) it is significant according to a t-test, and (2) it and the other lagged values of ΔX jointly add explanatory power to the model according to an F-test. Then the null hypothesis of no Granger causality is accepted if and only if no lagged values of ΔX have been retained in the regression.

## Description by Clive Granger

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, X_t, Y_t, and W_t, and that we first attempt to forecast X_{t+1} using past terms of X_t and W_t. We then try to forecast X_{t+1} using past terms of X_t, Y_t, and W_t. 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 X_{t+1} that is not in past X_t or W_t. In particular, W_t could be a vector of possible explanatory variables. Thus, Y_t would "Granger cause" X_{t+1} if (a) Y_t occurs before X_{t+1}; and (b) it contains information useful in forecasting X_{t+1} that is not found in a group of other appropriate variables.

Naturally, the larger W_t is, and the more carefully its contents are selected, the more stringent a criterion Y_t is passing. Eventually, Y_tmight seem to contain unique information about X_{t+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 Y_t to cause X_{t+1} and for X_t to cause Y_{t+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 ). 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.

Further references for this personal account are (Granger 1980; Granger 2001).

Limitations and Extensions Linearity The original formulation of G-causality can only give information about linear features of signals. Extensions to nonlinear cases now exist, however these extensions can be more difficult to use in practice and their statistical properties are less well understood. In the approach of Freiwald et al. (1999) the globally nonlinear data is divided into a locally linear neighborhoods (see also Chen et al. 2004), whereas Ancona et al. (2004) used a radial basis function method to perform a global nonlinear regression. Stationarity The application of G-causality assumes that the analyzed signals are covariance stationary. Non-stationary data can be treated by using a windowing technique (Hesse et al. 2003) assuming that sufficiently short windows of a non-stationary signal are locally stationary. A related approach takes advantage of the trial-by-trial nature of many neurophysiological experiments (Ding et al. 2000). In this approach, time series from different trials are treated as separate realizations of a non-stationary stochastic process with locally stationary segments. Dependence on observed variables A general comment about all implementations of G-causality is that they depend entirely on the appropriate selection of variables. Obviously, causal factors that are not incorporated into the regression model cannot be represented in the output. Thus, G-causality should not be interpreted as directly reflecting physical causal chains (see Personal account).