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!
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 &amp;= B_0^{-1} B_1 y_{t-1} + B_0^{-1} u_t \\
    &amp;= 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.)