# Covariance Between $\hat{\beta_0}$ and $\hat{\beta_1}$ [duplicate]

Our model is $Y=\beta_0+\beta_1X+U$.

We know that $$\hat{\beta_0} = \beta_0 + \sum\limits_{n=1}^N c_nu_n \quad \text{ and }\quad \hat{\beta_1} = \beta_1 + \sum\limits_{n=1}^N k_nu_n \,,$$ where $$k_n = \frac{(x_n-\bar{X})}{D}\,, \quad D = \sum_{n=1}^N (x_n-\bar{X})^2\,,\quad\text{ and }\quad c_n = \Big[\frac{1}{N}-\bar{X}k_n\Big].$$

We are trying to use the assumptions of autocorrelation and homoskedasticity to derive the covariance between the OLS slope and intercept estimators, $\mathrm{Cov}[\hat{\beta_0},\hat{\beta_1}]$.