Is this an endogeneity problem?

Question

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 $m$.

Answer 1

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

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

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.