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)?   
 A: 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.


*

*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.

*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.
A: 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: 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.
