OLS with Correlated Errors

Consider the standard linear regression model $y = X\beta + \varepsilon$, where OLS assumes $\text{Cov}(\varepsilon) = \sigma^2 I$. Now suppose the errors are not i.i.d. but instead exhibit correlation. 1. If the errors are **positively correlated**, what happens to the OLS coefficient estimates $\hat{\beta}$? Are they still unbiased? What about the estimated standard errors and the resulting t-statistics? 2. If the errors are **negatively correlated**, how does the story change? 3. What would you do in practice to fix these issues?