I am trying to test if $a_t$ granger causes $b_t$:
$$a_t=\rho_1+\phi_{11}a_{t-1}+\alpha_{11}b_{t-1}+\beta_{11}c_{t-1}+u_{1t}$$
$$b_t=\rho_2+\phi_{21}a_{t-1}+\alpha_{21}b_{t-1}+\beta_{21}c_{t-1}+u_{2t}$$
$$c_t=\rho_3+\phi_{31}a_{t-1}+\alpha_{31}b_{t-1}+\phi_{31}c_{t-1}+u_{3t}$$
I am trying to use an F test.
However, should be null be: $\phi_{21}$ = 0.
Or do I also need to set: $\beta_{21}$ = 0 because a impacts b through c.