Sufficient condition of the Gauss-Markov Theorem

Question

For a simple linear regression model under the assumptions of the Gauss-Markov theorem, for the least squares estimators of intercept and slop to be uncorrelated, does the sample average of the dependent variable has to be zero?

Answer 1

Yes. Assume the following model $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, where $\mathbb{E}=0$, $cov(\epsilon_i, \epsilon_j)=0$, $i\neq j$ and $var(\epsilon_i)=\sigma^2 <\infty$. So the OLS estimators are given by $\hat{\beta} = (X'X)^{-1}X'y$, and $cov(\hat{\beta})=\sigma^2(X'X)^{-1}$, i.e., $cov(\hat{\beta}_0, \hat{\beta}_1)=cov(\hat{\beta})_{12}=cov(\hat{\beta})_{21}$, where $$ (X'X)^{-1}= \frac{1}{n\sum(x_i - \bar{x}_n)^2} \begin{pmatrix} \sum x_i ^2 && \ -n\bar{x}_n\\ -n\bar{x}_n && n \\ \end{pmatrix}, $$ thus if $\bar{x}_n=0$ then $\rho = \frac{cov(\hat{\beta}_0,\hat{\beta}_1)}{\sigma_{\hat{\beta}_0}\sigma_{\hat{\beta}_1}}=0$.