Identities in a VAR model I am working on a VAR model where one of the equations is an identity. For example:
$$ \begin{cases}
A_t = \alpha_{11} + \alpha_{12} A_{t-1} + \alpha_{13} B_{t-1} + \alpha_{14} C_t + \varepsilon^{(1)}_t\\
B_t = \alpha_{21} + \alpha_{22} A_{t-1} + \alpha_{23} B_{t-1} + \alpha_{24} C_t + \varepsilon^{(2)}_t\\
C_t = A_t + B_t
\end{cases}$$
Where $A_t$, $B_t$, and $C_t$ are time series variables, $\alpha$'s are the coefficients to be estimated, and $\varepsilon$'s are the error terms. 
So far, my idea is to estimate the first two equations, then generate fitted values $\hat{A}_t$ and $\hat{B}_t$, and finally to use $\hat{A}_t + \hat{B}_t$ as an estimate of $C_t$. However, this procedure does not take into account the identity (the last equation) when estimating the first two equations. Are there any methods (or ways to transform the model) that would actually incorporate the information from the identity into the estimation procedure? 
 A: Your setup seems an little bit unusual for me, especially given that you are introducing simultaneity/endogeneity in the model by adding the contemporaneous variable C. This means actually you are dealing with a Structural VAR (SVAR), not a simple VAR. Note you can rewrite this as:
$$ \begin{cases}
A_t = \alpha_{11} + \alpha_{12} A_{t-1} + \alpha_{13} B_{t-1} + \alpha_{14} (A_t+B_t) + \varepsilon^{(1)}_t\\
B_t = \alpha_{21} + \alpha_{22} A_{t-1} + \alpha_{23} B_{t-1} + \alpha_{24} (A_t+B_t) + \varepsilon^{(2)}_t
\end{cases}$$
and hence: 
$$\left(\begin{array}{cc}
1-\alpha_{14}&\alpha_{14}\\
\alpha_{24}&1-\alpha_{24}
\end{array}\right)
\left(\begin{array}{c}
A_{t}\\B_t
\end{array}\right)=
\begin{array}{l} \alpha_{11} + \alpha_{12} A_{t-1} + \alpha_{13} B_{t-1}  + \varepsilon^{(1)}_t\\
 \alpha_{21} + \alpha_{22} A_{t-1} + \alpha_{23} B_{t-1} + \varepsilon^{(2)}_t
\end{array}$$
You could then eventually pre-multiply the right hand side by the inverse of the left matrix to get a reduced form and hence a standard VAR. But would face two issues:


*

*The errors of the reduced form will be correlated, even if you assumed the errors in the structural form weren't. 

*Recovering your structural parameters, in particular the $\alpha_{14}$, $\alpha_{24}$, will be difficult, given the specific restriction you have. 


Alternatively, look whether you find restricted estimators for the "SVAR" structural form that can take into account your specific restrictions. 
