Consider the basic form of a linear regression: $y_i= a+b_1x_i+u_i$
Two of the most important assumptions in OLS is the conditional mean zero: $E[u_i|x_i]=E[u_i]=0$ and the know as "orthogonality in mean": $E[u_ix_i]=0$. In many cases these two assumptions are problematic, and for that reason I was wondering about the implications of non-compliance each of them. I know that the conditional mean zero is fundamental to show that the beta estimation under OLS is unbiased and for that reason is important this assumption but I do not know how non-compliance of the orthogonality can affect the estimation by OLS.
Thanks in advance.