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Why is the assumption $E[\varepsilon X]=0$ weaker than $E[\varepsilon\mid X]=0$?

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I'm just looking from a probabilistic perspective, and looking forward to hear other ideas. Using law of iterated expectations, we have $$E[\epsilon X]=E[E[\epsilon X\mid X]]=E[X E[\epsilon\mid X]]$$ If $E[\epsilon\mid X]$ is $0,$ then automatically, $E[\epsilon X]$ is $0.$ But, the other way around is not true, which means $E[\epsilon X]$ is a weaker condition.

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    $\begingroup$ Indeed (+1). In fact $E(u\mid X) = 0 \implies E[f(X)u] = 0$ for any function $f$. $\endgroup$ – Alecos Papadopoulos Nov 4 '18 at 19:35

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