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In simple regression model, the zero conditional mean assumption implies u and x are uncorrelated as a slighter condition of zero conditional mean assumption. Then they use this to estimate. But I don't know why zero conditional mean assumption implies this after reading couple of books. Someone can explain?

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If $E[u|x]=0$, then $$E[ux]=E[E[ux|x]]=E[xE[u|x]]=0$$ which means $\operatorname{cov}(u,x)=E[ux]-E[u]E[x]=0$ because also we have $$E[u]=E[E[u|x]]=E[0]=0$$

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    $\begingroup$ Why you put x outside of the expected value if it is a random variable? $\endgroup$
    – robertspierre
    Commented Apr 30, 2023 at 3:21
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    $\begingroup$ If you mean in the first line, i.e. $E[ux|x]$, $x$ is given so it can be removed from the inside of the expectation. $\endgroup$
    – gunes
    Commented Apr 30, 2023 at 7:14

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