We say that in "standard" OLS regression the residuals $û$ are uncorrelated with $k$-th explanatory variable $x_k$. I know the argument can be intuitively derived from geometry of OLS. There are a lot of interesting articles in this forum on that topic with the bottom line, that the projection is orthogonal.
I am only interested, if that is also true in the following three cases:
- we do a regression without a constant
- $x_k$ is not exogenous
- $E(x_{ik}' u_i) = 0$
I am not quite sure about the first one. I think I've heard, that we need the constant in the model for it to be true. The exogeneity should not matter, as far as I know, i.e. the projection is still orthogonal. And I think the third one does not matter for the question.