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Consider the following model:$$y=x\beta+u$$

Now, let us take conditional expectations, assuming conditional exogeneity, such that $$E[u|x]=0$$ $$E[y|x]=\beta E[x|x]=\beta x$$

By the Law of Iterated Expectations:$$E_{x}[E[y|x]]=E[y]=E_{x}[\beta x]$$

What does the last term simplify to?

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  • $\begingroup$ E[y]? I'm not sure why you re-wrote E[y] as the last term. They are equivalent, but doesn't E[y] provide more insight? $\endgroup$
    – tjnel
    Dec 5, 2015 at 17:25

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Switching the last two terms makes more sense. For $E[y|x]$ you plug in $\beta x$, then the whole term can be simplified to $E[y]$, which is the result. $E[y]$ is the weighted average (w.r.t. $x$) of $E[y|x]$.

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