Why do we use Vector Autoregressive Models? Let's say we want to estimate the system
$x_{1,t}=\phi_0+\phi_1 x_{1,t-1}+\phi_2 x_{2,t-1} +\epsilon_t$
$x_{2,t}=\gamma_0+\gamma_1 x_{1,t-1}+\gamma_2 x_{2,t-1} +\eta_t$
Do we gain anything be estimating using a VAR method in terms of consistency, standard errors of the estimates, power of the tests, etc. as opposed to simply estimating the equations separately using appropriate time series techniques?
 A: Your question can be addressed under the Seemingly Unrelated Regression (SUR) theory. Normally if $\epsilon_t$ and $\eta_s$ are not correlated other than when $t=s$ the SUR theory suggests that there is no difference between estimating the equations separately or alltogether as long as all equations contain the same explanatory variables (which in this case they do).
You may read the whole text or as well jump to the last page.
http://www.phdeconomics.sssup.it/documents/Lesson17.pdf
However when there is correlation not only contemporaneously (at the same time point) but across time I believe this result doesn't apply anymore. In this situation a GLS estimation of the equation bundle may be more efficient. But I don't have a reference at hand to back up my claim at the moment. Nevertheless since in VAR models across time correlation is assumed to be zero (https://en.wikipedia.org/wiki/Vector_autoregression)  this situation shouldn't be much of a concern. 
A: Trying a short non-technical answer. Univariate time series models like AR (or ARMA) for stationary time series try to autoproject the future of the series from its past. Thas is, your projections only comes from the history of the series. That may be fine, maybe you do not have any other information on other variables.  But maybe you have some other time series (measured at the same points in time), as an example inflation, wages and unemployment. Now, certainly, say wages probably do depend some on unemployment, with low unemployment the employers are competing for workers and need to pay more. So, maybe, the time dynamics of these series do involve all three at once, and if that is true we might make better forecasts of, say, unemployment if we take into account also wages and inflation. That lead to VAR models. 
To see if it is worth it, build first some univariate time series models and calculate some measure of prediction quality. Then you can use that as a baseline for your VAR model. 
I am sure other reasons can be given, econometricians use such models to test hypothesis about the economy, but I have no experience with such use.
