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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:

  1. What would be wrong with simply estimating the system by OLS (equation by equation)?

  2. 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.

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Predictions in a system in which (almost) all regressors are endogenous

If you are interested in prediction only, endogeneity is not the core problem, while overfitting is. Read here: Endogeneity in forecasting

Moreover

What would be wrong with simply estimating the system by OLS (equation by equation)?

If you are focusing on prediction only, equation by equation is good; systems can be avoided.

Your second question

Is there an alternative approach that allows us to study how the u's affect the $x$'s?

seems more focused on causal questions. Like manageable with Structural VARs

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  • $\begingroup$ Thanks! +1 Your link which in turn is linked to Francis Diebold's discussion about T-consistency and P-constistency is extremely helpful. $\endgroup$ – Bert Breitenfelder Jan 7 at 8:00

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