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I have a variable $A$ which Granger causes a variable $B$, and a variable $C$, separately.

Since A and B are very similar in nature, I should expect that $B$ granger caused $C$ as well. Anyway when I perform the test I can't reject the null hypothesis of no causality between them.

Am I missing by any chance something about the theory and the "math" behind the test?

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    $\begingroup$ Why do you think this is true? And can you come up with a counter-example where it is false? Also read the wiki on the self-study tag and figure out if adding it is appropriate. $\endgroup$
    – Matthew Gunn
    Commented Nov 5, 2016 at 6:35
  • $\begingroup$ Since the statement (1) "$A$ Granger-causes $B$ and $C$" refers to $B$ and $C$ symmetrically--they can be interchanged without changing the meaning at all--then if you could conclude "$B$ Granger-causes $C$" then you could also conclude "$C$ Granger-causes $B$". These two statements allow you logically to deduce that $B$ and $C$ "Granger-cause" each other. (Note that the actual meaning of "Granger-causes" is irrelevant to this argument.) Does it make sense that statement (1) could imply such a conclusion? $\endgroup$
    – whuber ♦
    Commented Nov 5, 2016 at 21:55
  • $\begingroup$ I'm sorry, I have miswritten my question. I meant that A causes B, and A causes C but separately. In other words, I perform two different test in which I verify the causality between and A and B and between A and C. $\endgroup$
    – James
    Commented Nov 8, 2016 at 21:00
  • $\begingroup$ I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? $\endgroup$
    – Richard Hardy
    Commented Feb 24, 2017 at 14:17

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Since this is a self-study question, I will give you some hints rather than a full answer.

To falsify a conjecture, you only need to find a counterexample. Let us see how easy (or difficult) it is.

  • You can formulate Granger noncausality in terms of restrictions in a VAR model.
  • Take the simplest VAR model -- a VAR(1) -- for a trivariate system and see what minimal restrictions are sufficient for $A \not\xrightarrow{Granger} \{B,C\}$.
  • Consider whether these restrictions imply $B \not\xrightarrow{Granger} C$.
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  • $\begingroup$ I'm sorry but I haven't made myself clear in my first question. I did not mean that I have a trivariate system. $\endgroup$
    – James
    Commented Nov 8, 2016 at 21:03
  • $\begingroup$ @James, well, this actually does not change the essence and the answer still holds. $\endgroup$
    – Richard Hardy
    Commented Nov 9, 2016 at 6:15
  • $\begingroup$ @James, so, is the answer clear? Do you need another hint? $\endgroup$
    – Richard Hardy
    Commented Nov 14, 2016 at 18:45

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