I'm trying to finish a proof for a review exercise and I'm asked to show that
where $y$ is the dependent variable and $f(x)$ is a linear predictor of $y$.
I'm almost finished, but I just want to check whether or not
$E\left[ E \left[ y E(y|x)\,|\,x \right] \right]=E\left[ E(y|x)E \left[ E(y|x)\,|\,x \right] \right]$
Basically, can I take $y$ out of the conditional expectation as $E(y|x)$, or worded differently, are conditional expectations multiplacative in this way?
Ordinarily you can back a function out of the expectation if it is a function of the variable being conditioned on - i.e., $E\left[ f(x)y|x \right]=f(x)E(y|x)$, but I think what I'm trying to do above is different.
Also, a second, related question – if $E\left[E(y|x)\right]=E(y)$ by the Law of Total Expectations, then does $E\left[E(y|x)E(y|x)\right]=E(y^2)$ by the same token?