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Consider the general problem of predicting the conditional mean $E(Y|X)$ where $X$ is the predictor. One assumes $Y$ can be written as:

$Y=f(x)+e$ where $E(e|X)=0$ which implies covariance of predictor and $e$ is 0. Is $E(e|X)=0$ an assumption that is made or is it something that follows automatically from the fact that $f$ is the conditional mean of $Y$ (conditioned on $X$)?

If it's an assumption, what drives us to make it?

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  • $\begingroup$ I can't figure out what your "$x$" refers to: I'll assume it means "$X.$" You seem to be asking that if $f(X)= E[Y\mid X] = E[f(X)+e\mid X],$ then can we conclude $E[e\mid X] = 0$? If so, then look up linearity of expectation and taking out what is known. As far as "what drives us," could you tell us what kind of alternative you are contemplating? $\endgroup$
    – whuber ♦
    Mar 14 at 23:05
  • $\begingroup$ Understood, thanks! I was trying to split conceptually a conditional expectation and causality in my head. They are not the same thing is what I concluded. $\endgroup$
    – Arshdeep
    Mar 17 at 13:18

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