What's the deal with VARs? I'm new to econometrics/statistics. Why are Vector Autoregressive (VAR) equations considered a separate class of models in textbooks, apart from ARs? Isn't it true that you can estimate the equations one by one and arrive at the same estimates compared to estimating the equations simultaneously? Why even call them VARs instead of saying "we have multiple AR models"?
 A: First of all, multiple AR would normally mean that only own lags are included as regressors in each equation. Meanwhile, lags of other endogenous variables are included in addition to own lags in a VAR.
Regarding estimation: technically, you may indeed estimate VAR using equation-by-equation OLS. However, it need not always be the efficient way to do it. Generally, a feasible GLS estimation is the efficient one.
One notable exception where feasible GLS does not improve on equation-by-equation OLS is when the set of regressors (the right hand side variables) are the same in each equation in the system. Another exception is when the error terms are uncorrelated across equations.
Most of the textbooks covering VAR models consider these issues and provide a more detailed discussion then here.
A: They are vector autoregressions because the residual covariance matrix and the coefficient covariance matrix can have (but does not always have) relationships among residuals/coefficients of different equations. Actually, this is something you can input into your prior, or parameterize your likelihood function to take these cross-equation relations into account.
