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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.

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This site https://www.statisticshowto.com/granger-causality/ shows about conducting the F-test for granger-causality. It also talks about alternative Tests.

"If you have a large number of variables and lags, your F-test can lose power.

An alternative would be to run a chi-square test, constructed with likelihood ratio or Wald tests. Although both versions give practically the same result, the F-test is much easier to run."

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  • $\begingroup$ Links are great, but bringing a summary of the the contents into the answer would make it even better. That way the answer can stand on its own, and in the event that the website disappears your answer can remain a valuable resource. $\endgroup$
    – BKay
    Oct 19, 2020 at 14:07

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