# Help me get my head around this simple linear regression problem

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?

• 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. Feb 10 at 15:46

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