# Granger Causality with or without VAR

I would like to test two stationary time series (say a and b) for Granger Causality. I am familiar with the method of running two regressions:

a=const+a[-1]+e_1
a=const+a[-1]+b[-1]+e_1


And then using an F Test (I omitted the degrees of freedom): (R2f-R2r)/(1-R2r) Afterwards I do the same the other way around:

b=const+b[-1]e_2
b=const+b[-1]+a[-1]+e_2


This would then tell me which of them Granger causes the other one.

How is a VAR different from this (or is the set up above basically already a VAR)?

From the perspective of testing for Granger causality, your setup is exactly the same as a VAR(1) setup if equation-by-equation OLS is used for estimating the restricted VAR.

1. The coefficient estimates and their standard errors in your individual equations will be obtained in exactly the same way as in the VAR(1) model and will have the same values, both in the restricted case and the unrestricted case.
2. The test will be carried out in the same way: the $F$-statistic will be calculated from the same elements and it will be compared to the same critical values.

Thus I see no difference between your approach and a VAR(1) when it comes to testing for Granger causality.

• That is exactly what I thought, thank you so much Richard!! – Niccola Tartaglia Jun 23 '17 at 15:12
• @NiccolaTartaglia, You are welcome! – Richard Hardy Jun 23 '17 at 15:22
• It's more like you're saying that from a Granger causality test setup one can build a VAR. But for the F-test you need the error sum of squares from the restricted model, which you don't get from a standard reduced VAR estimation. – Lucas Farias Jun 23 '17 at 20:45
• @lucasfariaslf, thanks for the comment! What I am saying is the other way around: if you have a VAR(1) and the OPs equations and want to test for GC, then there is no difference which one you use. You are right that you do not get sum of squares of the restricted model from standard VAR estimation; that is why I wrote explicitly in my point 1. that there is the restricted and the unrestricted case. I also wrote in my first paragraph that there is a restricted VAR model being estimated. Does that explain it? – Richard Hardy Jun 24 '17 at 8:06
• @RichardHardy with all these considerations, yes. Still, seems like we had this possible "equivalence" of the procedures thought in different ways, that is way, to me, answering OP's question with no further assumptions leads to a negative answer. – Lucas Farias Jun 25 '17 at 19:13

The VAR is a model, and OLS is a technique to estimate a model, but in essence, yes, that is it - one typically estimates VAR models with OLS regressions for each equation and assesses the significance of each of the coefficients on the lags of the variables of the variable whose Granger causality is to be investigated.

This happens more commonly with a $\chi^2$-test that is asymptotically valid even without normal errors, unlike the F-test, but that distinction is often not practically important.

A vector autoregression (VAR) intends to model existing linear relationships in set of time series. On the other hand, Granger-causality is a concept (related to the relevance of other variables in forecasting another one), that can be tested, for two variables (for example, since it can be multivariate), using the linear models you cited.

Specifically, the set up in a Granger-causality test you presented do not correspond to a VAR, since the latter must contain - in its reduced form - lags of both variables in a symmetric fashion. For example, a VAR(1) on your variables $a$ and $b$, would look like:

$$a_t=c_a+\beta_1^a a_{t-1}+\beta^a_2b_{t-1}+e^a_t\\ b_t=c_b+\beta_1^b b_{t-1}+\beta^b_2a_{t-1}+e^b_t$$

The system above together with some assumptions regarding the error terms completely defines a VAR(1) model. Now compare this with the set up you presented and the difference should be better illustrated.

• Granger causality can be tested for two or more variables and between two or more groups of variables, not just two variables. – Richard Hardy Jun 22 '17 at 7:52
• @RichardHardy That's why I left it explicit in my first paragraph I was addressing the bivariate case, actually for making it easy to compare with a VAR(1). Still, the idea of the explanation doesn't change for more than two variable. – Lucas Farias Jun 22 '17 at 12:01
• It just appeared to me that there is a flavour of contrast between your first and second sentences, and I found it misleading. Not to say you are wrong, but perhaps it can be phrased in a way such that the apparent contrast is avoided? – Richard Hardy Jun 22 '17 at 12:22
• What is still confusing me a bit however is that fact Granger causaility appears to be examined within VARs. From our discussion so far, my understanding is that I could analyze Granger causality without a VAR using my approach above (for 2 or more variables). But I could also do this using a VAR, but I would not have to. So if I chose to use a VAR for other purposes, I could just test Granger causality along the way and I would get the same result as I would from my approach above, right? – Niccola Tartaglia Jun 22 '17 at 14:49
• Put differently, in this context a VAR is just one method of conducting a Granger causality test (which should yield the same result as my method above). Of course a VAR also serves a whole number of other purposes ... – Niccola Tartaglia Jun 22 '17 at 14:53