Consider the following system where all variables are endogenous.
\begin{align*} x_{1} & =\beta_{21}x_{2}+\beta_{31}x_{3}+u_{1}\\ x_{2} & =\beta_{12}x_{1}+\beta_{32}x_{3}+u_{2}\\ x_{3} & =\beta_{13}x_{1}+\beta_{23}x_{2}+u_{3}% \end{align*}
The $u$'s are error terms. There are no instruments available. With all regressors endogenous, the $\beta$'s will be biased when estimated by OLS, however, I am only interested in prediction. That is, I need to know how the variables change, when the $u$'s change.
Rewriting the system in matrix notation and solving for the $x$ vector gives
\begin{align*} \left( \begin{array} [c]{c}% x_{1}\\ x_{2}\\ x_{3}% \end{array} \right) =\left( \left( \begin{array} [c]{ccc}% 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right) -\left( \begin{array} [c]{ccc}% 0 & \beta_{21} & \beta_{31}\\ \beta_{12} & 0 & \beta_{32}\\ \beta_{13} & \beta_{23} & 0 \end{array} \right) \right) ^{-1}\left( \begin{array} [c]{c}% u_{1}\\ u_{2}\\ u_{3}% \end{array} \right) \end{align*}
Questions:
What would be wrong with simply estimating the system by OLS (equation by equation)?
Is there an alternative approach that allows us to study how the $u$'s affect the $x$'s?
Edit: I tried to keep the question as simple as possible, but now I think I have to give more information. There is a follow-up question asking about inference. The follow-up question also gives some more context.
For each $x_{k}$ there are $T\times S$ observations ($S$ is the number of species and $T$ is days). When estimating by OLS (for each $k$ separately), the regression equation is $$ x_{k,s,t}=\sum_{k=1,k\not =l}\beta_{lk}x_{l,s,t}+\delta_{k,s}+\varepsilon _{k,s,t}% $$ where $\delta_{k,s}$ is a $k$-$s$-fixed-effect and $\varepsilon_{k,s,t}$ the residual.
