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I am studying linear regressions. In this business, sometimes we can prove the results we want with the assumption that the error term $U$ is such that $E(XU) = 0$. But for lots of other results, in special the ones that require a causal interpretation, we need to assume $E(U|X) = 0$.

I understand how the latter implies the former. But I am struggling with an example where the first is verified while the second is not. Can anyone provide some intuition/counter-example? Thanks a lot in advance!

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Consider $X$ uniform in $[-1,1]$ and $U = |X|$. Would you allow that as a counter-example, even though very artificial?

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