I saw the following statement in my lecture note:
"The data generation process is $y = x+\epsilon$, whereas in the regression we run y on x so the regression model is $y = \beta_{OLS} x+e$, the approximation error is $e=y-\frac{E[yx]}{E[x^2]}x$. Note that for the joint gaussian distribution of x and y, we have $\rho x=E\left [ \epsilon|x \right ]$, where $\rho = 1-\beta_{OLS}$"
I feel $\rho x=E\left [ \epsilon|x \right ]$ comes from $E\left [ e|x \right ]=0$, but I'm not sure how we can reach this?
Does anyone understand what is the logic here? Thanks a lot!