Structural VAR and Granger Causality

Question

Is it possible to Structural VAR (vector autoregression) model to imply Granger Causality? In other words, if X and Y are determined at the same time, is it possible that X Granger-causes Y? Thanks!

Answer 1

Every structural VAR (SVAR) model, e.g. $$ B_0 y_t = B_1 y_{t-1} + u_t $$ has an equivalent reduced form (VAR), e.g. \begin{aligned} y_t &= B_0^{-1} B_1 y_{t-1} + B_0^{-1} u_t \\ &= A_1 y_{t-1} + \varepsilon_t. \end{aligned} where $A_1 := B_0^{-1} B_1$ and $\varepsilon_t := B_0^{-1} u_t$.

The reduced form can be used directly for testing Granger causality. In the example, one would test whether certain off-diagonal elements in $A_1$ are equal to zero. E.g. if $y_t$ is bivariate, then testing $H_0\colon a_{12}=0$ would tell us whether $y_2$ does not Granger-cause $y_1$. The structural form might even imply Granger non-causality if it implies that certain off-diagonal elements are zero.

However, Granger non-causality does not preclude finding an equivalent representation of the reduced form that has a nondiagonal $B_0$ matrix. In fact, you would just multiply any reduced form (whether Granger-causal or not) by any nondiagonal matrix, and that matrix would then be your $B_0$. Hence, $B_0$ being nondiagonal is not informative of presence or absence of Granger causality. (This is a sort of tautology, I think.)

Answer 2

The nature of the Granger causality test is that determines if another series past values preforms a better job at forecasting the series of interest than the series of Interest's own values.

If X and Y are procyclical and move together in the same wave form throughout time they would not be good predictor of future values.

i.e. Increase in Investment and Consumption would not "Granger Cause" GDP since they are contemporaneous terms.