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We know that the best 1st order approximation of an unconditional expectation is the following-

$$E(y|x)=(E(y)-\beta E(x))+\beta x$$

where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}=\frac{E(xy)-E(x)E(y)}{E(x^2)-[E(x)]^2}$.

Is there an a general expression available that would expand the above to include the higher orders of $x$ analogous to the Taylor's Series Expansion?

$$E(y|x)=a_0+a_1x+a_2x^2+...\infty$$

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  • $\begingroup$ What do you want to do with this? What properties to you expect from the remainder? The kind of "higher order" expansion that I have found occasionally useful are Hoeffding decompositions. $\endgroup$ – Andreas Dzemski Aug 20 '18 at 12:00
  • $\begingroup$ It would allow for a representation of the coefficients of the regression $$y_t=\epsilon_t+a_0+a_1x_t+a_2x_t^2+...$$ I am interested in seeing how the addition of the higher order terms affects the value of the coefficients. e.g. If we assume that $E(y|x)$ is constant, then $a_0=E(y)$. If $E(y|x)=a_0+a_1x$, then $a_0=E(y)-\beta E(x)$ and so on. $\endgroup$ – Amrit Prasad Aug 21 '18 at 2:33

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