I would like to know if the logic in these two situations is correct.
Situation 1: Let's say we have a continuous dependent variable, $y_1$, that then has a causal impact on an unobserved variable, $\rho$. This unobserved variable then has a causal impact on a variable, $y_2$, which has a causal impact on $y_1$. We want to estimate $ \frac{\partial y_1}{\partial y_2}$. So, we have a set of structural equations as follows:
$$y_1 = f(y_2, \mathbf{x_1})+ e_1$$ $$\rho = f(y_1, \mathbf{x_2})+ e_2$$ $$y_2 = f(\rho, \mathbf{x_3})+ e_3$$
where the $\mathbf{x_i}$ terms are exogenous and the $e_i$ terms are errors. By substituting the second equation into the third, we can see that we would have simultaneity and our estimates of the impact of $y_2$ on $y_1$ would be biased if we could not control for $\rho$ or some proxy for it in our estimate of the structural equation for $y_1$.
I am particularly unsure about this last part in italics. Could we use the proxy for $\rho$ in the estimate of the structural equation for $y_1$, or would we have to do 2SLS, with the proxy for $\rho$ being included in the first stage but excluded in the second?
Situation 2: Let's say we have a percentage dependent variable, $s_1$. Let's say the complement of $s_1$ is made up of two other percentages, $s_2$ and $s_3$. Furthermore, let's say that $s_3$ has a causal impact on an unobserved factor, $\rho$, and that $\rho$ has a causal impact on $y_2$, which has a causal impact on $s_1$. We want to estimate $\frac{\partial s_1}{\partial y_2}$. Thus, we have the following structural equations:
$$s_1 = f(y_2, \mathbf{x_1})+ e_1$$ $$\rho = f(s_3, \mathbf{x_2})+ e_2$$ $$y_2 = f(\rho, \mathbf{x_3})+ e_3$$
Let's now say that $s_2$ is more or less constant across observations. Thus, there is generally an inverse relation between $s_1$ and $s_3$. This implies that we can rewrite $s_3$ in the second structural equations in terms of $s_1$:
$$\rho = f(1 - (s_1 + \bar{s}_2), \mathbf{x_2})+ e_2$$
Then, just as in situation 1, by substituting this equation into the structural equation for $y_2$, we can see there would be simultaneity and our estimates of the impact of $y_2$ on $y_1$ would be biased (once again, I am unsure about whether controlling for $\rho$ or a proxy for it in our estimate of the structural equation for $y_1$ would solve this).