In regression, the residuals should be what?

• A. Uncorrelated
• B. Correlated
• C. Normally distributed
• D. Homoscedastic

Answer: (A)

Residuals are the differences between what the model predicts and what was actually observed. After accounting for the predictor(s), there shouldn’t be any systematic pattern left in those errors with respect to the predictors. When residuals are uncorrelated with the predictor variable(s), it means the model has captured the relationship with X and there’s no remaining linear association driving the errors, which supports the validity of the estimated effect of X.

Normal distribution of residuals and constant variance (homoscedasticity) are useful for reliable inference (t-tests, confidence intervals) but are not required for the coefficients to be unbiased. You can have non-normal or heteroscedastic residuals and still have correct estimates, though inference becomes more delicate.