universe in which is excluded. If the above hypothesis is accepted, we say that Granger-causes . Method If a time series is a stationary process, the test is performed using the level values of two (or more) variables. If the variables are non-stationary, then the test is done using first (or higher) differences. The number of lags to be included is usually chosen using an information criterion, such as the Akaike information criterion or the Schwarz information criterion. Any particular lagged value of one of the variables is retained in the regression if (1) it is significant according to a t-test, and (2) it and the other lagged values of the variable jointly add explanatory power to the model according to an F-test. Then the null hypothesis of no Granger causality is not rejected if and only if no lagged values of an explanatory variable have been retained in the regression. In practice it may be found that neither variable Granger-causes the other, or that each of the two variables Granger-causes the other. 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 an 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. Multivariate analysis Multivariate Granger causality analysis is usually performed by fitting a vector autoregressive model (VAR) to the time series. In particular, let for be a -dimensional multivariate time series. Granger causality is performed by fitting a VAR model with time lags as follows: where is a white Gaussian random vector, and is a matrix for every . A time series is called a Granger cause of another time series , if at least one of the elements for is significantly