Consider the following simultaneous system
$$ y_{1} =\beta _{1}y_{2}+\alpha _{1}z+u_{1} \\ y_{2} =\beta _{2}y_{1}+\alpha _{2}z+u_{2} $$
where $y_{1}$, $y_{2}$ and $z$ are vectors of random variables each of length $n$. $u_{1}$ and $u_{2}$ are residuals and all greek letters are coefficients to be estimated.
The Wikipedia entry on the simultaneous equation model mentions the possibility to estimate all simultaneous equations at once. For large systems, this will be computationally costly but for smaller ones this seems a reasonable alternative.
Question: What is the best way to estimate (by least squares) the system at once?
Crossposting: A couple of days ago, I have posted the same question on stats/stackexchange.com. So far, the question has not received any answers or comments there.