The least squares estimator of $\beta_0$ is given as $\hat{\beta_0}=\overline{Y}-\hat{\beta_1}\overline{X}$. Find Cov $\big(\hat{\beta_0},\hat{\beta_1}\big)$. When is Cov $\big(\hat{\beta_0},\hat{\beta_1}\big)=0$?
I'm not sure how to solve this. I started out by plugging in $\hat{\beta_0}$ into the Covariance to try and solve for $\hat\beta_1$. Can anyone help me out with this??