# Does the Granger Causality test in the “vars” package make sense?

We all understand that the Granger Causality test entails constructing two models. The first one is simply an autoregressive model with $y_{t-1}$ being the single independent variable. The second one is adding an $x_{t-1}$ to the autoregressive model. Next, you check whether the residuals of those two models are different using an $F$-test. If they are, you can say that $x$ Granger-causes $y$.

However, within the "vars" package in R, when you use the causality function it works differently. That's because you can only specify what is $x$. The package's author calls it a "cause" variable. But you can't specify what is $y$, or the response variable. So when the R output comes out, you get that $x$ Granger-causes several different variables simultaneously with one single given $p$-value.

Within the framework of Granger Causality as depicted above, this does not seem to make much sense. I can understand variable $A$ Granger-causes variable $B$. But I don't understand how variable $A$ Granger-causes variables $B$, $C$, $D$ with a single $p$-value estimate just as if you had done $A$ Granger-causes $B$ alone.

• Can you post a reproducible example? My guess would be it means "A causes at least one of B, C, D." – Christoph Hanck Dec 21 '15 at 4:46
• Christoph, I am attempting to copy the output I have copied from R onto an Excel spreadsheet. I don't know if I'll have much success with that. Granger-causality testing Personal Income granger causing H6DDA growth. > causality(var3, cause = "pi", vcov. = NULL, boot = FALSE, boot.runs=100) $Granger Granger causality H0: pi do not Granger-cause h6dda ten data: VAR object var3 F-Test = 5.7548, df1 = 4, df2 = 606, p-value = 0.0001501 – Sympa Dec 21 '15 at 19:39 • The most important section of the R output above is: "Granger causality HO: pi do not Granger-cause h6dda ten. pi stands for Personal Income, h6dda is Deposit (time series DDA with H6 money aggregates) and ten stands for 10 Year Treasury. In other words, this Granger-causality test tests whether Personal income (the cause in the coding of the R function) Granger-cause Deposit growth and also 10 Year Treasury movements... which does not make much sense to me. – Sympa Dec 21 '15 at 19:45 • Please edit your original question - this is fairly hard to read. Also, this is not reproducible, just the output. – Christoph Hanck Dec 22 '15 at 8:51 • Someone else already edited my question. I am not too sure how to edit it further. Given the issue, it is reasonably well described. The issue in a nutshell is very simple. We all understand how Variable A Granger-causes Variable B (I have described the mechanics of Granger Causality testing above). But, what does Variable A Granger-causes Variables B, C, and D at the same time mean? I argue that there is a bug in this causality() function. And, it is missing an argument. You should be able to select the variable that is affected. – Sympa Dec 22 '15 at 18:11 ## 1 Answer When I first used the causality function of the vars package I had the same doubt. Here is what I thought. Imagine a trivariate VAR(1) model: $$Y_t = a_0 + a_1 Y_{t-1} + a_2 X_{t-1} + a_3 Z_{t-1} + \epsilon_{y,t} \\ X_t = b_0 + b_1 Y_{t-1} + b_2 X_{t-1} + b_3 Z_{t-1} + \epsilon_{x,t} \\ Z_t = c_0 + c_1 Y_{t-1} + c_2 X_{t-1} + c_3 Z_{t-1} + \epsilon_{z,t}$$ In order to$X_t$not Granger cause$Y_t$you need to make sure that$H_0: a_2 = a_3 = 0$or$H_0: a_2 = c_2 = 0$. You can build an$F$-test for that, but what the causality function seems to be doing is checking if$X_t$Granger causes all other variables in the model. Why would they do that? Maybe because it is simpler. That would be the same as to just check if$H_0: a_2 = c_2 = 0$. This is a much more simple$F$-test, and it extends easily to more than three variables. So, if I am correct, this is not a bug, they simple took the easiest path. So if the null is not rejected, this means that$X_t$does not Granger cause$Y_t$and$Z_t$. But if the null is rejected, the test doesn't say much,$X_t$may be Granger causing either$Y_t$,$Z_t$or both. Note that in the help of the causality function they only show a bivariate case, but from that example you can infer that the trivariate case would be as I described. To make sure that this is the case, one can build the corresponding$F\$-test and check if the values are equal. Also, if you want to specify the response variable you can use the grangertest function of the lmtest package.

• Regis, thanks that's a pretty good answer. I am not entirely comfortable with the Granger causality framework as depicted. It would seem way too noisy. I actually really like your suggestion of using granger test () within the lmtest package. That most probably implements Granger causality in a well established fashion. – Sympa Jan 26 '16 at 22:27