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I'm studying Econometrics, and I'm trying to understand the endogeneity by simultaneity problem in the OLS estimation. Given a generic OLS model: $$y = \underline{x} \underline{\beta} + u$$ where $\underline{x}$ is the vector of explanatory variables. My professor gave the classical, and a bit vague, explanation of simultaneity saying that it happens when an explanatory variable $x_i$ and $y$ are "codetermined", without specifying exacly what her meant by this term. I looked up the definition of simultaneity given in this wikipedia page.

In the wikipedia page it says that

Suppose that two variables are codetermined, with each affecting the other. Suppose that there are two "structural" equations, $$\begin{align*} y_i &= \beta_i x_i +\gamma_i z_i + u_i \\ z_i &= \beta_2 x_i + \gamma_2 y_i + v_i \end{align*} $$ [...] Assuming that $1-\gamma_1 \gamma_2 \neq 0$ and that $x_i, v_i$ are uncorrelated with $u_i$ then $$E(z_i u_i) = \dots \neq 0$$ Therefore there is endogeneity.

Now, I want to truly understand what it means for two random variables (in this case $z_i$ and $y_i$) to be codetermined. To me, this concept is highly related to the uncorrelation of the two error terms and the condition on the $\gamma$ coefficients, in fact, given any structural equation, I can always isolate an explanatory variable and write it as a linear combination of the other explanatory variables and $y_i$, but generally in this case both the condition on uncorrelation of errors and the condition on $\gamma$ will be violated, of course. So can someone explain to be, in rigorous probabilistic terms, what it means for two random variables to be codetermined (and what simultaneity is truly about)? Thanks

EDIT: I'm a math grad student, and this is the first econometric course that I'm taking, so I cannot but use a mathematical statistics approach. My question is a meta-econometric question, in the sense that the all the usual explanations I've seen (although intuitive) of this phenomenon are not precise. My question revolves around the mathematical relation between the two random variables ($y_i$ and $z_i$ in the example). For example:
1) Does codetermination means that the random variable $z_i$ is $y_i$-measurable and vice-versa? This would imply that we can write both of them as a (deterministic) function of the other: $$z_i = f(z_i)$$ $$y_i = g(z_i)$$

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    $\begingroup$ your system of equations in the highlighted block is pretty much a definition of linear simultaneous equations model. so, what's your difficulty here? $\endgroup$ – Aksakal Oct 14 at 18:53
  • $\begingroup$ Perhaps you will find this example illuminating. $\endgroup$ – Dimitriy V. Masterov Oct 14 at 18:56
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The simultaneity induces simultaneous equations bias. It seems you're fine with this link, and simply want a formal definition of simultaneous equations model to start with. It can be found in multiple econometrics textbooks. Here's a snippet from the chapter in Greene's 7th edition:

enter image description here

The system 10-42 above is the definition of simultaneous equations model. You can find all the details in the textbook, of course. You can recognize the system in your question as a special case of Eq.10-42

I suppose it's called simultaneous because you have to determine all $y_i$ at once, not one after another. However, it's more nuanced than that. For instance, seemingly unrelated regressions look quite similar to this model (except no endogenous variables on the right side), and they don't have the same endogeneity issues.

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