# Is conditional mean independence, $E(e_i | x_i)=0$, a different assumption from $E(x e) = 0$? (from Hansen)

This question comes out of Hansen's Econometrics ((https://www.ssc.wisc.edu/~bhansen/econometrics/Econometrics.pdf))

In section 2.18, we only impose the assumptions of finite variance and $$Q_{xx}$$ being positive definite, and then derive the linear projection coefficient $$\beta = E(\textbf{x} \textbf{x}^\prime)^{-1} E(\textbf{x} y)$$ as the minimizer of the expected squared error of the linear projection model $$y = \textbf{x}^\prime {\beta} + e$$. This leads to the implication (NOT the assumption) that $$E(\textbf{x} e) = \textbf{0}$$.

Then in section 4.4, under assumption 4.2, we have: $$E(e_i | \textbf{x}_i)=0$$.

My question is: is $$E(e_i | \textbf{x}_i)=0$$ a newly-imposed assumption in chapter 4? Or is it equivalent to the condition $$E(\textbf{x} e) = \textbf{0}$$ which we reached in chapter 2, not by assumption but by implication?

I know that $$E(\textbf{x} e) = E(E(\textbf{x} e|\textbf{x})) = E(\textbf{x} E(e|\textbf{x}))$$ (by law of iterated expectations and conditioning theorem, respectively), so imposing the assumption that $$E(e|\textbf{x}) = \textbf{0}$$ yields the implication that $$E(\textbf{x} e) = \textbf{0}$$. But is the reverse true? Does $$E(e_i | \textbf{x}_i)=0$$ follow from $$E(\textbf{x} e) = \textbf{0}$$, or is $$E(e_i | \textbf{x}_i)=0$$ a new assumption that we're imposing?

$$E(\mathbf{x}e)=0$$ does not imply $$E(e|\mathbf{x})=0$$, to see that, suppose that x is a variable that always takes the value of zero, but $$e$$ is independent of $$\mathbf x$$ and has a mean of, say, $$3$$. Then the expected value of the product is necessarily $$0$$, but the conditional expectation will be equal to 3 (given independence, the conditional mean is equal to the unconditional mean).