Theoretical concept of adding constraints to a VAR? I was wondering whether someone can explain why one would add constraints to a VAR model. Recommendations to books / articles explaining these concepts in detail are greatly appreciated.
 A: The starting point is, why would one expect that an unrestricted VAR model is necessarily a better approximation to a data generating process than a restricted VAR model?
It is certainly possible to think of real-world feedback systems between variables where feedback goes only one way (example 1 below) or there is a delay in response (example 2), or the amount of delay in response differs across VAR model equations. 
I will give examples of zero restrictions, but the general argument applies to other kinds of linear or nonlinear restrictions.
Example 1
Suppose that the true data generating process for a bivariate time series $(y_t, x_t)$ is 
$$ \begin{align}
 y_t &= a_{1,11} y_{t-1}              + a_{1,12} x_{t-1} + u_t \qquad (1) \\
 x_t &= \phantom{{}=a_{1,11} y_{t-1}}   a_{1,22} x_{t-1} + v_t \qquad (2) 
\end{align} $$
(The subscript of $a_{i,jk}$ reads as follows: $i$ stands for lag, $j$ and $k$ index the row and column of the corresponding coefficient matrix $A$ if the model is written in matrix notation.)
This could be considered a restricted VAR(1) model and written as
$$ \begin{align}
 y_t &= a_{1,11} y_{t-1}  + a_{1,12} x_{t-1} + u_t \qquad (1) \\
 x_t &= 0 \ \cdot \ \ y_{t-1} + a_{1,22} x_{t-1} + v_t \qquad (2a) 
\end{align} $$
In this system, $x$ is exogenous with respect to $y$. A restricted VAR model reflects the situation when the feedback between two variables goes one way but not both ways. This would not be possible in an unrestricted VAR model.
Now suppose you want to test whether $y$ Granger-causes $x$. Then you estimate the unrestricted model as well as the restricted model and compare them. Thus estimation of a restricted VAR is also instrumental in Granger causality testing.
Example 2
Suppose that the true data generating process for a bivariate time series $(y_t, x_t)$ is 
$$ \begin{align}
 y_t &= a_{2,11} y_{t-2} + a_{2,12} x_{t-2} + u_t \qquad (3) \\
 x_t &= a_{2,21} y_{t-2} + a_{2,22} x_{t-2} + v_t \qquad (4) 
\end{align} $$
This is a restricted VAR(2) model with $a_{1,11}=a_{1,12}=a_{1,21}=a_{1,22}=0$ (the whole first lag is missing). This happens if there is a delay in response. 
Alternatively, one could have different lags in the equations for $y_t$ and in the equation for $x_t$. That could be a VAR(2) model with $a_{1,11}=a_{1,12}=a_{2,21}=a_{2,22}=0$.
After all, the main point is that some processes are better approximated by a restricted VAR model, and I do not see a good reason for using an unrestricted VAR model in such situations.
Edit: as pointed out by @GraemeWalsh, restrictions may be of two types: restrictions on model coefficients (such as the ones discussed above) and restrictions on the contemporaneous covariance matrix of model errors. The latter would be considered in the context of structural vector autoregressions (SVAR) which I have limited experience with; I will not try my luck in expanding on it further.
