How does Granger Causality relate to Vector Autoregression? I'm curious how Granger causality plays into vector autoregression modeling. I've seen very technical explanations online, but a less confusing one would be very helpful.  Thanks.
 A: VARs are used to model the mutual relationship between variables. Let's consider the simple case of two variables like the quarterly-GDP growth and QuarterlyReturns of the S&P500 Index. 
In VAR, we want to analyze how the two variables (and their past realizations) influence each other, taking into account also any contemporaneous effect between the two. 
In our example, we assume a VAR(1), our model would look like: 
$ GDP_{t} = \gamma_{10} -  b_{12}SP_{t} + \gamma_{11}SP_{t-1} +\gamma_{12}GDP_{t-1} + e_{t}^{GDP}$
$ SP_{t} = \gamma_{20} -  b_{21}GDP_{t} + \gamma_{21}SP_{t-1} +\gamma_{22}GDP_{t-1} + e_{t}^{SP}$
NOTE: This is referred to as the Structural for of the VAR. Unfortunately, the model in such form cannot be estimated due to the presence of a contemporaneous effect ($SP_{t}$ in equation one and $GDP_{t}$ in equation 2) which makes OLS estimate unfeasible. We say that the model is not identified. To solve this there are many ways, such as Cholesky Triangularization that leads to the Reduced Form VAR from which the Structural can be retrieved by imposing some restrictions on the parameters. Nevertheless, we will keep it in this form since it's easier to interpret it. 
The two main purposes of our analysis are:
1. Use our model to forecast  $GDP_{t+k}$ and $SP_{t+k}$ with $k=1,2,3..$
2. Investigate the relationship between $GDP_{t+k}$ and $SP_{t+k}$
Granger Causality comes into play in 2., along with Variance Decomposition and Impulse Response Function. 
Blandly speaking, the idea of Granger Causality is to verify if $GDP$'s past value is 'useful' to predict $SP$ or if adding it to our prediction model for $SP$ does not increase the accuracy of the prediction, and vice-versa. 
In formal terms, it means comparing the 'accuracy' of the following two models:
$ SP_{t} = \gamma_{20} + \gamma_{21}SP_{t-1} +\gamma_{22}GDP_{t-1} + e_{t}^{SP}$
$ SP_{t} = \gamma_{20} + \gamma_{21}SP_{t-1} + e_{t}^{SP}$
If regression 1 is statistically better than regression 2, it means that the term $\gamma_{22}GDP_{t-1}$ does help to forecast $SP$, hence we say $GDP$ Granger-Cause $SP$ and we write $GDP_{t} \Rightarrow^{GC} SP_{t}$. 
In more formal terms, what we really do is a Likelihood Ratio Test or a F-Test on the two regression, where we want to statistically test the null hypothesis that the coefficient $\gamma_{22}$ related to the $GDP_{t}$ (in our case is only one lag term, but may be more than one depending on the lag order of the VAR model that we selected)
Please note that the word Causality is a misnomer. In fact, what Granger-Sims causality really means is a correlation between the current value of one variable ($SP_{t}$) and the past value(s) of another (other) variable(s) ($GDP_{t-1}$).
A final note. Usually one tests Granger Causality in both directions, yielding one of the following cases: 
$GDP_{t} \Rightarrow^{GC} SP_{t}$  but  $SP_{t} \not\Rightarrow^{GC} GDP_{t}$ SP IS EXOGENOUS TO GDP and GDP Granger-causes SP
$SP_{t} \Rightarrow^{GC} GDP_{t}$  but  $GDP_{t} \not\Rightarrow^{GC} SP_{t}$ GDP IS EXOGENOUS TO SP and SP Granger-causes GDP
$SP_{t} \Rightarrow^{GC} GDP_{t}$  but  $GDP_{t}\Rightarrow^{GC} SP_{t}$ SP and GDP are a FEEDBACK LOOP
$SP_{t} \not\Rightarrow^{GC} GDP_{t}$  but  $GDP_{t} \not\Rightarrow^{GC} SP_{t}$ SP and GDP are Linearly Unrelated
