For a general linear regression , are Y and Y hat independent?

Question

For a general linear regression , are $Y$ and $\hat{Y}$ independent?

$Y$=XB+e

$\hat{Y}$=X$\hat{B}$

I think they are dependent, because if they rely on the same data, they should have some sort of relationship.

that is to say , $\hat{Y}$=HY , so they are dependent right?

Answer 1

As long as B is not equal to 0, the answer is no. If B is not 0, then both terms will depend on X and so they cannot be independent.

$Cov(Y,\hat{Y})=Cov(XB+e,Ŷ) = B*cov(X,X\hat{b})$

Edit: I suppose if your model is misspecified, you could get an incorrect estimate of $\hat{b}=0$, and thus no correlation, but I don't think thats what you had in mind.