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Let's say we want to determine the impact of $y_2$ on $y_1$, which are related as follows:

$$y_1 = f(y_2, x_1,e)$$

where

$$e = g(y_3,x_2, u_1), \space y_2 = h(y_4,x_3, u_2), \space y_3 = m(y_4, x_4,u_3)$$

and the $u$'s are the exogenous errors. Substituting, we have

$$y_1 = f(h(y_4,x_3, u_2),x_1, g(m(y_4, x_4,u_3),x_2, u_1))$$

We can see here that there is a correlation between $h$ and $g$ in $f$. Therefore, if we do not control for $y_3$ or $y_4$ in $f$, we would have endogeneity because then $h$ would be correlated with the error. Is this correct?

In other words, we are looking for $\frac{\partial f}{\partial h}$, but a change in $h$ without holding $y_3$ or $y_4$ constant in $g$ implies that our estimate of $\frac{\partial f}{\partial h}$ is not ceteris paribus; when we adjust $h$, we also adjust $g$.

My main concern here is whether it matters that the correlation between $y_2$ and the error in $f$ occurs through two functions: $g$ and $m$.

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I think expressing your Structural Causal Model as Causal Graph gives simple answer. The model looks like:

enter image description here

Much clearer view we can obtain removing non-confounding exogenous variables (however this step is not necessary):

enter image description here

We can clearly see here, that $y_4$ is the confounder which creates backdoor path from $y_2$ to $y_1$ through $y_3$ and $e$. In order to block that backdoor path we need to condition for any nonempty set of the variables: $y_4$, $y_3$, $e$ (which I guess is unobservable error).

Therefore: yes, there exist endogeneity problem with identification of the causal effect because of existing confounder. The estimator in the model $y_1 = f(y_2, x_1, e)$ will most likely be biased even if $f$ is assumed correctly.

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  • $\begingroup$ Excellent. You confirmed my intuition, as I drew a very similar graph for myself. Now, the question I have is, what would the next step be to resolve this issue? If $y_3$ and $y_4$ are not directly observable and there are no good proxies for them, I would say, at least in theory, a good next step would be finding an instrument for $y_2$. $\endgroup$ – Ryan da Silva Mar 3 at 1:05
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    $\begingroup$ Yes, Instrument, (natural) experiment, sole mediator (front-door criterion) or measuring variables that let you close backdoor paths. I do not see any other options, however there are tons of natural-experimental designs. $\endgroup$ – cure Mar 3 at 9:32
  • $\begingroup$ Ok, but what if we could control for $y_3$ but not $y_4$? Wouldn't that solve the endogeneity issue? $\endgroup$ – Ryan da Silva Mar 4 at 5:58
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    $\begingroup$ Yes, controlling for a mediator blocks backdoor path. As I stated in the answer, any nonempty set of those three variables leads to the correct identification of effect of $y_2$ on $y_1$. It could be any of those variables, or more of them. $\endgroup$ – cure Mar 4 at 17:12
  • $\begingroup$ Thanks for the help. If you have time, I have a related question here that I would love to hear your feedback on: stats.stackexchange.com/questions/512340/… $\endgroup$ – Ryan da Silva Mar 4 at 23:40

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