Granger causality test on residuals I have a time-series of three variables: $x_0, x_1, x_2$. I have a theory that claims $x_0$ granger causes both $x_1$ and $x_2$.
On the other side I want to check if $x_1$ causes $x_2$, $x_2$ causes $x_1$, or both.
As $x_0$ causes both other variables I though of running 2 different var models:

*

*$\textrm{var}(x_0,x_1)$

*$\textrm{var}(x_0,x_2)$
Then I get the residuals of $x_1$ and $x_2$ from these models and call them $rx_1$ and $rx_2$. Finally I apply the following var model: $\textrm{var}(rx_1, rx_1)$, after which I test for Granger causality. In other words, as I know that $x_0$ causes both $x_1$ and $x_2$ to vary, therefore, I control for $x_0$ and then check if the residuals, $rx_1$ and $rx_1$, granger cause each other.
My question would be: Does this procedure make sense (statistically)? Am I controlling the effect of $x_0$? And if so, is there a book chapter or article that introduces or uses a similar approach? I need to cite something in order to justify my procedure!
 A: Why are you not running a VAR model with all three variables in it? It is the sounder approach.
Regarding your idea, if

*

*$x_1\xrightarrow{Granger}x_2$ or $x_2\xrightarrow{Granger}x_1$ and

*lags of $x_0$ are correlated with lags of $x_1$ and $x_2$ and

*you run pairwise VARs for $(x_0,x_1)$ and $(x_0,x_2)$,

these pairwise VARs will capture some of the Granger causality of $x_1\xrightarrow{Granger}x_2$ or $x_2\xrightarrow{Granger}x_1$ in their fitted values, so their residuals will have less of it than there originally were. Therefore, testing for it using residuals form these pairwise VARs may yield weaker findings.
This is analogous to omitted variable bias in regression. If the true model is $$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\varepsilon$$ where $\text{Corr}(x_1,x_2)\neq 0$, you cannot recover $\beta_2$ by first running a regression of $y$ on $x_1$ and then its residuals on $x_2$. This is because in the first regression the effect of $x_2$ on $y$ is partly captured by $x_1$ due to its correlation with $x_2$, so there is less of it left in the residuals.
