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Given a very simple linear projection model $Y=X\beta+e$ with $E(e|X)=0$ and $X$ a scalar. Notice that this is a simple linear model with no intercept.

Then $\beta = E(XY)/E(X^2)$ from the least squares formula.

On the other hand, if I take expectation on both sides of the original model, I get $E(Y)=\beta E(X)+E(e)=\beta E(X)$. Therefore, $\beta=E(Y)/E(X)$. (Of course, suppose $E(X)\neq 0$.)

I just don't see how the two quantities could be identical. Did I do something wrong?

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  • $\begingroup$ If the model were to assume a non-zero intercept, $\alpha$, then the OLS regression results in a estimate for $\alpha$ given by $$\hat \alpha =E(Y)-\hat \beta E(X)$$From this, note that a zero value of $\hat \alpha$ would correspond with an estimate for $\beta$ given by $$\hat \beta =E(Y)/E(X)$$ So, this is consistent with your result. $\endgroup$
    – Mark Viola
    Feb 10, 2021 at 15:46

1 Answer 1

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$E[f(X)Y]=E[f(X)X]\beta+E[f(X)e]=E[f(X)X]\beta$, so $\beta=\frac{E[f(X)Y]}{E[f(X)X]}=\frac{E[Y]}{E[X]}=E[\frac{Y}{X}]=E[\frac{f(X)}{E[f(X)]}\frac{Y}{X}]$ ...

Strange ...

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  • $\begingroup$ Sorry I don’t follow. Is this an answer? $\endgroup$
    – Bunbury
    Feb 10, 2021 at 15:27
  • $\begingroup$ I think so, but I'm not sure how to interpret it. $\endgroup$
    – Ian Zhang
    Feb 10, 2021 at 15:30

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