Prove that a random error and the fitted value of y are independent

Question

I have a simple linear regression $y=\beta_{0}+\beta_{1}x + \epsilon$ and the formula for the fitted regression $\hat y=\hat \beta_{0}+\hat \beta_{1}x$. The book takes for granted that the following is true $$\Bbb E[\epsilon \hat y]=\Bbb E[\epsilon]\Bbb E[\hat y]$$ and that means that $\epsilon,\hat y$ are independent random variables. Can you show me why this is true?

Answer 1

$\hat{Y}$ is projection of $Y$ onto $C(X)$, hence just show that $$ \mathbb{E}[\epsilon X]=0. $$ Namely, $$ \mathbb{E}[\epsilon X]=\mathbb{E}[Y - \beta_0 - \beta_1x |x] = \mathbb{E}[Y\mid x] - \mathbb{E}[\beta_0 + \beta_1x\mid x] $$ $$ \mathbb{E}[\epsilon X]=\mathbb{E}[\beta_0 + \beta_1x\mid x]-\mathbb{E}[\beta_0 + \beta_1x\mid x]=0. $$ I.e., it shows that $\operatorname{cov}(X,\epsilon)=0$, thus as $\hat{Y}$ is linear projection, $\hat{Y} = HY = f(X)$, $\operatorname{cov}(f(X),\epsilon)=0$.