implications of non-compliance with the assumptions in the estimation of beta by ols

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.

• You might find it helpful to contemplate an extreme example of failure, such as $x_i=u_i.$ In that case the OLS solution is already in front of you and you can compare it with the truth. Note, too, that $E[u_i\mid x_i]=0$ already implies $E[u_ix_i]=E[E[u_i\mid x_i]x_i]=E[0(x_i)]=0.$
• Oh I see thats a very elegant way in order to see that conditional mean zero implies orthogonality. I understood, but suppose that the conditional mean zero is not satisfied so if the orthogonality in mean also do not satisfied, how this can affect the OLS estimation? I think that is probably that the $x_i$ have to be necessary correlated to the error and that is a problem but in specific for OLS idk Jun 23 at 21:47