# In a predictive model, is orthogonality of noise and predictors an assumption or something that can be proved?

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?

• 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?
– whuber
Mar 14 at 23:05
• 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. Mar 17 at 13:18