# How to test for Granger causality?

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.