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  • $\begingroup$ Hi: If $E(U_{i}) \neq 0$, then the model has an error term with a non-zero mean. Intuitively, that would make the $\beta$ estimate not consistent and your equations bear that intuition out. $\endgroup$ Commented Jul 28, 2022 at 19:31
  • $\begingroup$ Why? Please can you show it formally? I never use $E(U_i)=0$ in my proof. This is indeed the point I make in my question! $\endgroup$ Commented Jul 28, 2022 at 19:35
  • $\begingroup$ Hi TEX: You use $E(X_{i}U_{i}) = 0$ in your proof and I assume that that is based on the orthogonality assumption which is based on $E(U_i) = 0$. Is it not ? If it is, then, if you don't have $E(U_i) = 0$, then you won't have orthogonality so that last term in your last equation won't be 0 ? $\endgroup$ Commented Jul 29, 2022 at 3:38
  • $\begingroup$ Why is the orthogonality assumption based on $E(U_i)=0$? Could you explain? It does not seem so if I read Hayashi p.112 google.com/url?sa=t&source=web&rct=j&url=https://… $\endgroup$ Commented Jul 29, 2022 at 4:18
  • $\begingroup$ Also read Hayashi p.109 $\endgroup$ Commented Jul 29, 2022 at 4:23