Can we assume that the correlation of the residuals $û$ from an OLS regression with the explanatory variables is always equal to zero?
The orthogonal projection matrix is given as $P = X (X'X)^{-1} X'$ and does not involve any assumptions about the the error term or the residuals as far as I know. We only need full rank of $X$ in order to invert $X'X$.
But came across some answers here on stackexchange, stating that we need a constant in the model for it to be true. In other words we would need $E[u_i]=0$ for uncorrelatedness.
If I put together the informations correctly, I think it all comes down, whether orthogonality and uncorrelatedness are the same?
By now, I would say, that orthogonality always holds assuming full rank, while for uncorrelatedness we need the constant in the model. This also implies that orthogonality and uncorrelatedness do not automatically imply each other. But maybe someone can shed some more precise light on this.