In simple linear regression, which of the following is a residual independence assumption?

In simple linear regression, which of the following is a residual independence assumption?

• A. Normality of residuals
• B. Linearity
• C. Homoscedasticity
• D. Independence of residuals

Answer: (D)

The key idea here is that the residuals—the differences between what the model predicts and what is actually observed—should not be correlated with each other. When residuals are independent, knowing one residual gives no information about another, which lets us trust the standard errors and test statistics we compute for the regression coefficients. This independence is separate from how the residuals are distributed (normality), whether the relationship is truly linear (linearity), or whether the residuals have constant variance across the range of X (homoscedasticity). Normality affects inference in small samples, linearity ensures the model form is appropriate, and homoscedasticity affects the precision of estimates; independence specifically guards against bias in the standard errors that would arise if residuals were correlated. If residuals were correlated (autocorrelation), we’d need to adjust our methods, for example with robust standard errors or a different modeling approach.

The key idea here is that the residuals—the differences between what the model predicts and what is actually observed—should not be correlated with each other. When residuals are independent, knowing one residual gives no information about another, which lets us trust the standard errors and test statistics we compute for the regression coefficients. This independence is separate from how the residuals are distributed (normality), whether the relationship is truly linear (linearity), or whether the residuals have constant variance across the range of X (homoscedasticity). Normality affects inference in small samples, linearity ensures the model form is appropriate, and homoscedasticity affects the precision of estimates; independence specifically guards against bias in the standard errors that would arise if residuals were correlated. If residuals were correlated (autocorrelation), we’d need to adjust our methods, for example with robust standard errors or a different modeling approach.