If error terms are correlated between observations, will that reduce how predictive a model is? Specifically, given the same predictor variables, will the mean square error of the model's predictions tend to be larger when the correlation between error terms in larger?
I understand how it increases the standard error of the coefficients of predictors, but I don't understand how it increases the uncertainty about predictions of a dependent variable. For example, suppose that I wanted the predict future values of some outcome. A simple prediction could be to guess the average. I don't see how correlated errors would reduce how accurate that prediction is. You're still getting another sample of the value of the outcome.
Why does it reduce predictive power, if it does? What is a simple example of why correlated error terms reduce predictive accuracy?
Would accounting for the correlation improve the model accuracy? How would one remove the correlation?