Granger Causality

Econometrician Clive Granger (1969) has developed a widely used definition of CAUSALITY. A key aspect of the concept of Granger causality is that there is no instantaneous causation (i.e., all causal processes involve time lags). Xt “Granger causes” Yt if information regarding the history of Xt can improve the ability to predict Yt beyond a prediction made using the history of Yt alone. Granger causality is not unidirectional; that is, Yt Granger causes Xt if information about the history of Yt can improve the ability to predict Xt beyond a prediction made using only the history of Xt.

Granger causality tests may be performed using a bivariate vector autoregression (VAR) model. A VAR for X and Y specifies that each variable is a function of k lags of itself plus k lags of the other variable; that is,

An F-test or a Lagrange multiplier (LM) test can be used to determine if X Granger causes Y or Y Granger causes X. For example, the LM test to see if X Granger causes Y involves estimating (1) under the assumption that the λs are jointly zero. Residuals from this regression then are regressed on all variables in (1). The R2 from this second regression ×T (number of observations) is the test statistic (distributed as χ2, df = 2k + 1). Rejection of the null hypothesis implies X Granger causes Y. An alternative VAR representation of the above test has been developed by Sims (1972). The impact of nonstationarity on the validity of these tests is controversial. Granger causality tests also may be implemented in the ARIMA modeling framework (see Freeman, 1983).

The concept of Granger causality has been incorporated into the definitions of exogeneity offered by Hendry and associates (e.g., Charemza & Deadman, 1997, chap. 7). Recognize that one need not establish that X Granger causes Y to warrant inference concerning the effects of X in a single-equation model of Y. It is sufficient to establish that X is weakly exogenous to Y (i.e., parameters in the model for Y can be efficiently estimated without any information in a model for X). However, if one wishes to forecast Y, then it is necessary to investigate if X is strongly exogenous to Y. Strong exogeneity equals weak exogeneity plus Granger “noncausality” (i.e., Yt does not affect X at time t or at any time t + k). If Y is not strongly exogenous to X, then feedback from Y to X must be taken into account should that feedback occur within the forecast horizon of interest.