Which statement correctly contrasts homoscedasticity and heteroscedasticity?

Which statement correctly contrasts homoscedasticity and heteroscedasticity?

• A. Homoscedasticity means constant residual variance across levels of X; heteroscedasticity means residual variance changes with X
• B. Homoscedasticity means residuals are normally distributed; heteroscedasticity means residuals are not
• C. Homoscedasticity means independence of residuals; heteroscedasticity means residuals are correlated
• D. Homoscedasticity means residual variance increases with X; heteroscedasticity means constant variance

Answer: (A)

Homoscedasticity means the residual variance is the same across all values of the predictor(s). In a good regression model, the spread of the residuals should be roughly constant no matter where you are on X. When the residuals fan out or tighten as X increases (the spread changes with X), that’s heteroscedasticity. This distinction matters because constant variance supports valid standard errors and hypothesis tests in ordinary least squares. If the variance of the residuals changes with X, the standard errors can be biased, which can lead to unreliable inferences even though the estimated relationships might still be unbiased. The other statements mix in different ideas. Normal distribution of residuals concerns the shape of the distribution, not whether the spread is constant across X. Independence of residuals refers to whether residuals are uncorrelated with each other, a separate assumption. Saying variance increases with X describes heteroscedasticity, not homoscedasticity.

Homoscedasticity means the residual variance is the same across all values of the predictor(s). In a good regression model, the spread of the residuals should be roughly constant no matter where you are on X. When the residuals fan out or tighten as X increases (the spread changes with X), that’s heteroscedasticity.

This distinction matters because constant variance supports valid standard errors and hypothesis tests in ordinary least squares. If the variance of the residuals changes with X, the standard errors can be biased, which can lead to unreliable inferences even though the estimated relationships might still be unbiased.

The other statements mix in different ideas. Normal distribution of residuals concerns the shape of the distribution, not whether the spread is constant across X. Independence of residuals refers to whether residuals are uncorrelated with each other, a separate assumption. Saying variance increases with X describes heteroscedasticity, not homoscedasticity.