# What is the difference between $E[\varepsilon\mid X]=0$ and $E[\varepsilon X]=0$ in OLS regression?

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

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