# Looking for definition: Granger Causality with a common explanatory process

I am looking for help in defining my problem.

Essentially, I have two data processes (both continuous macroeconomic variables: $x$ and $y$). There is evidence of bidirectional causality between $x$ and $y$, proven by a Granger Causality test.

What I am alleging now is that $x$ and $y$ are not causing each other, but that they are both caused by a third process, $z$, which is the reason for the causation between $x$ and $y$. So, $x$ and $y$ do not actually cause each other, but they are both simply caused by $z$.

I'm thinking about it this way: $x$ and $y$ appear to GC each other, since some events of $x$ precede events of $y$ and vice versa. But if there is a process $z$, whose events always precede those of $x$ and $y$, than $x$ and $y$ do not necessarily cause each other.

For now, I am trying to frame this problem properly, i.e. I am looking to define it. I could not find any literature on this particular type of problem, but I am sure there must be some. I assume that I'm simply not searching for the correct terms. Hence, my (hopefully) simple question: How is this phenomenon called?

(I would of course appreciate any suggestions on literature as well)

What you are looking is probably tests for block exogeneity. Granger causality has a nice interpretation in terms of certain coefficients in VAR model being zero. Suppose we have the VAR model for $(x_t,y_t)$:

\begin{align} \begin{bmatrix} x_t\\ y_t \end{bmatrix}=\begin{bmatrix} c_1\\ c_2 \end{bmatrix} +A_1 \begin{bmatrix} x_{t-1}\\ y_{t-1} \end{bmatrix}+...+ A_p\begin{bmatrix} x_{t-p}\\ y_{t-p} \end{bmatrix} +\begin{bmatrix} \varepsilon_{t}\\ \varepsilon_{t} \end{bmatrix} \end{align}

Then $y$ fails to Granger cause $x$ if matrices $A_1,...,A_p$ are lower triangular matrices, i.e. the upper right elements are zero. This concept easily generalises to three variables. In your case the $z$ equation in the VAR system should have only the lags of $z$, and these lags should be present in $x$ and $y$ equations. Then both $x$ and $y$ will fail to Granger cause $z$, but not vice-versa.

Technical details are probably more complicated, but the I hope I managed to convey general idea. You can read about this in Hamilton's book (J. D. Hamilton, Time Series Analysis, page 309), or you can search Google with key words Granger block exogeneity tests.

• Thanks for your answer. I looked up block causality in Enders' "Applied Econometric Time Series". That does seem to be the appropriate way to look at this problem. I am still puzzled of how you would call the process $z$, however. Since it plays a very particular role in a Var of $x$, $y$ and $z$, I would assume that there is a specific term to refer to it. Jul 2, 2014 at 15:09

I did a similar VAR model where my variables have bidirectional causality. What you ought to do is to look at the Variance Decomposition and assess how exogenous are your variables among each other.

Say, if x has .9 for its shocks and so does y, then z is most likely an omitted variable that you'd have to hypothesize using existing economic theory.

In which case, you'd have to define it in a separate section and cite some references.(Ideally, you'd search for data for this variable and rerun the model so you can correct bidirectionality)