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Suppose a linear predictor of the form $a + b'X$. To find estimators for a and b, should we minimize $E[Y-a-b'X]^2$ or $E[(Y-a-b'X)^2|X]$.

Former gives $\hat{a} = E[Y] - b'E[X]$ and latter gives $\hat{a} = E[Y|X] - b'X$

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You most likely want the former.

  • You probably don't want your $\hat{a}$ to depend on the value of $X$.
  • Even if you do want a dependence between the two, computing $E[Y|X]$ is something you would have to model again (possibly using a linear estimate) so you probably haven't gained much.
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