Given the simple linear regression $y_i = \beta_{i}x_i + \sigma\epsilon_i$, I need to find out the variance of residuals.
I have found out $\hat{\beta} = \frac{\sum_{i = 1}^{n}x_iy_i}{\sum_{i = 1}^{n}x_i^2}$ and $E[r_i] = 0$. I tried to apply the definition of residuals $r_i = y_i - \hat{y_i}$ and $ \hat{y_i} = x_i\hat{\beta}$. That is $Var(r_i) = Var(y_i - \hat{y}) = Var(y_i) + Var(\hat{y}) - 2Cov(y_i, \hat{y})$. But I have difficulty finding out $Cov(y_i, \hat{y})$