In a test of Granger causality, Christopher Sims exploits the fact that the future cannot cause the present.* To decide whether a variable Y causes a variable X, Sims suggests estimating the following pair of equations:

These regressions include the lagged, current, and future, or lead, values of the regressors; terms such as X_t+1, X_t+2, etc., are called lead terms.

If Y is to Granger-cause X, then there must be some relationship between Y and the lead, or future, values of X. Therefore, instead of testing that Σβ_i = 0, we should test Σλ_i = 0 in Eq. (1). If we reject this hypothesis, the causality then runs from Y to X, and not from X to Y, because the future cannot cause the present. Similar comments apply to Equation (2).

To carry out the Sims test, we estimate Eq. (1) without the lead terms (call it restricted regression) and then estimate Eq. (1) with the lead terms (call it unrestricted regression). Then we carry out the F test as indicated in Equation (8.7.9). If the F statistic is significant (say, at the 5% level), then we conclude that it is Y that Granger-causes X. Similar comments apply to Equation (2). Which test do we choose€”Granger or Sims? We can apply both tests.* The one factor that is in favor of the Granger test is that it uses fewer degrees of freedom because it does not use the lead terms. If the sample is not sufficiently large, we will have to use the Sims test cautiously.

Refer to the data given in Exercise 12.34. For pedagogical purposes, apply the Sims test of causality to determine whether it is sales that causes plant expenditure or vice versa. Use the last four years€™ data as the lead terms in your analysis.

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