In linear regression assumptions, what does independence of error terms mean?

In linear regression assumptions, what does independence of error terms mean?

• A. Error terms are uncorrelated with each other
• B. Error terms are always zero
• C. Error terms are independent of predictors
• D. Error terms are always positive

Answer: (A)

Independence of the error terms means the deviations for different observations do not move together. In practice this is expressed as the errors being uncorrelated across observations: Cov(ε_i, ε_j) = 0 for i ≠ j. If one observation has a larger-than-expected error, it doesn’t imply another will, so there’s no pattern linking the residuals from different cases. This lack of relationship helps ensure the ordinary least squares estimates have desirable statistical properties and that standard errors are valid for making inferences. The idea is not that errors are always zero—each observed outcome can deviate from the predicted line, but on average those deviations cancel out. Nor is it required that errors be completely independent of the predictors; the usual assumption is that the mean of the error is zero given the predictors (and, often, that errors are uncorrelated with each other). And errors are not restricted to be positive; they can be either sign.

Independence of the error terms means the deviations for different observations do not move together. In practice this is expressed as the errors being uncorrelated across observations: Cov(ε_i, ε_j) = 0 for i ≠ j. If one observation has a larger-than-expected error, it doesn’t imply another will, so there’s no pattern linking the residuals from different cases. This lack of relationship helps ensure the ordinary least squares estimates have desirable statistical properties and that standard errors are valid for making inferences.

The idea is not that errors are always zero—each observed outcome can deviate from the predicted line, but on average those deviations cancel out. Nor is it required that errors be completely independent of the predictors; the usual assumption is that the mean of the error is zero given the predictors (and, often, that errors are uncorrelated with each other). And errors are not restricted to be positive; they can be either sign.