Is this an endogeneity problem?

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 $$m$$.

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.

• 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 $y_2$. – Ryan da Silva Mar 3 at 1:05
• 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. – cure Mar 3 at 9:32
• Ok, but what if we could control for $y_3$ but not $y_4$? Wouldn't that solve the endogeneity issue? – Ryan da Silva Mar 4 at 5:58
• 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. – cure Mar 4 at 17:12
• 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/… – Ryan da Silva Mar 4 at 23:40