# 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 both arguments of $$f$$. Therefore, if we do not control for $$y_4$$ 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 $$y_4$$ 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 $$m$$.

• Is e your error term? Which of these variables you mention can you observe and which not? Mar 3 at 8:37