How can we determine the sign of Granger causality in a >2 dimensional VAR? As the title suggests, I'm trying to test for the sign of Granger causality in a large VAR. For exposition, consider the following three-dimensional VAR:
\begin{align}
\vec y_t=\vec c+\sum_{\ell=1}^pA_\ell \vec y_{t-\ell},
\end{align}
where $\vec y_t=(x_{1t},x_{2t},x_{3t})'$. I want to determine the sign of the causal relationship $x_1\leftarrow x_2$. My first thought was to use
\begin{align}
\text{sign}\left(\sum_{\ell=1}^p [A_\ell]_{12}\right)
\end{align}
and call it a day, but this ignores the impact of $x_3$ on the system.
I now know that the correct way to infer the sign is to use the impulse response function, based on the following comment from Dave Giles's blog. As implied by his comment, however, if the IRF is not (weakly) positive or negative, then the sign of the causality likely depends on the time horizon. This suggests to me that perhaps I should be referring to the impulse transfer function to infer how the sign of causality depends on frequency.
Is my thinking on this matter correct? And if so, how can I correctly use the impulse transfer function to determine how the sign of causality depends on frequency?
 A: There are many cases in analyzing multivariate systems:
2-Variable-Systems:


*

*In a 2-variable system where both of the variables are stationary, classical GC test [2] is achieved.   

*In a 2-variable system, if the variables are cointegrated (in this case, both are nonstationary), a vector autoregressive (VAR) model is not formed by differencing the variables (in this case, a vector error correction (VEC) model is formed) and G-noncausality tests are not performed via t/F tests (classical G-noncausality test) [5].   

*In a 2-variable system, when at least one of the variables is nonstationary, the G-causalities among variables is examined by Toda-Yamamoto G-noncausality test [6] since testing G-causality using F statistics (classical G-noncausality test) may result in spurious G-causality [7].  


[2] Granger, Clive W.; “Investigating Causal Relations by Econometric Models and Cross-spectral Methods”, Econometrica, vol. 37, no. 3, 1969, p. 424-438.
[5] Enders, Walter; Applied Econometric Times Series, 3.ed., Wiley, 2010, p. 321.
[6] Toda, Hiro Y.; Yamamoto, Taku; “Statistical Inference in Vector Autoregressions with Possibly Integrated Processes”, Journal of Econometrics, vol. 66, 1995.
[7] He, Zonglu; Maekawa, Koichi; “On Spurious Granger Causality”, Economics Letters, vol. 73, no. 3, 2001, p. 307-313.  
">2"-Variable-Systems:
When there are ">2" variables in a system upon which the G-causalities will be searched, then there is a risk of "spurious" G-causality, and hence the classical setup (Granger1969) cannot be used. Instead Advanced (Modern) GC techniques (Conditional GC, Partial GC, Global GC, etc.) are employed. First, digest the difference between the Classical GC (Granger1969) and Advanced (Modern) GC (Geweke1986, Ding2006, Guo2008, Seth2014).
Classical Granger Causality Analysis:
In a classical G-causality analysis, “multivariate” means there are more than 2 variables in the system or model in discussion for which G-causalities will be found. “Pairwise” means the G-causality analysis is performed by taking the variables in pairs (the number of the both of independent and dependent variable is just 1). For example, if the system’s variables are “x, y, z, w”, then “whether x G-causes y or not”, “whether x G-causes z or not”, etc. are analyzed without taking the effects of all the remaining variables in the system into account. Definitely, this latter is in fact not a G-causality: even though Granger first defined G-causality as the causality between 2 variables, he also emphasized that for finding G-causalities in systems with more than 2 variables, “all of the information in the system must be used for a G-causality analysis”. In search of “whether x G-causes y or not” that is isolated from the other variables, the effects of the variables of z and w are omitted and thereby some of the information in the system is not used. Hence, in essence, one can say that the pairwise G-causality of classical G-causality is in fact not a causality in the Granger sense!
Modern (Advanced) Granger Causality Analysis: In advanced G-causality analysis, “multivariate” means there are more than 1 variables in either of the “set of causers” (like-independents) and “set of caused by the causers” (like-dependents) in search of G-causality. Remember that in any Granger causality analysis, the variables are not pre-determined as independents and dependents; G-causality analysis reveals which ones are independent and which ones are dependent. For example, assume that there are 5 variables (x, y, z, q, w) in the system or model in which we search G-causalities among variables. Then, the (searched) G-causalities
•   “x and y G-causes z conditional on q and w”,                   (pattern: 5,2,1)
•   “x G-causes y and z conditional on q and w”,                   (pattern: 5,1,2)
•   “x and y G-causes z and q conditional on w”,                   (pattern: 5,2,2)
•   “x, y and z G-causes q and w”,                                         (pattern: 5,3,2)
•   “x and y  G-causes z, q and w”                                         (pattern: 5,2,3)
…
are all “multivariate” G-causalities in an advanced G-causality analysis. Notice pattern is “the number of all variables in the system, the number of variables in the causers set, the number of variables in the caused-by-the-causers set”. “Pairwise” means there are only 1 variable in both “set of causers” and “set of caused by the causers”. Hence, continuing with the above example, the (searched) G-causality of “x and y G-causes z conditional on q and w” is not a pairwise G-causality. On the other hand, “x G-causes y conditional on z, q and w”, “q G-causes z conditional on x, y and w”, … are all “pairwise” G-causalities.
Now, it is time for the answer of your question:
I think there are multiple ways to understand the sign, one may use the following:
Establish a VAR from the (arbitrarily ordered) variables in the dataframe of the system. Notice that as long as you do not dive on impulse-response function, forecast error variance decomposition etc. the order of the variables in the dataframe does not matter (the order of the variables in the impulse-response Cholesky definition in Eviews (2nd tab) corresponds to the order of the variables in the dataframe in R). Then, based on the significance of the coefficients in VAR regressions, deduce the sign.
Last advice: Study the Spurious GC, and then dwell on the GC in ">2"-variables-systems.
