My professor states the validity condition in the following way: $W$ (the matrix of instrumental variables) must be such that $ \text{plim} \, \frac{W'u}{N}=0$, where $u$ is the error term. Intuitively, I can see that in some way, this relates to the covariance between $W$ and $u$, but am not entirely convinced. Moreover, why do we look at this condition in an asymptotic setting? Why not impose mean independence, i.e., $E(u|W)=E(u)$?
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Essentially, $Cov(u,W) = 0$ is implied by $E(u|W) = E(u)$ by the Law of iterated expectations
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$\begingroup$ In general, uncorrelated random variables need not be mean independent. Unless you are talking about an asymptotic case when they are normal......? $\endgroup$– StudentDec 5, 2018 at 10:40
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1$\begingroup$ I think $\mathrm{Cov}(u,W)=0$ is implied by $E(u|W)=0$, not the other way around. $\endgroup$– Herr K.Dec 6, 2018 at 21:04
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