# Structural VAR and Granger Causality

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!

• Is my answer OK or do you need further clarification? I see you have not accepted any of the answers yet. Feb 3 '18 at 20:14

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.)