Which statement defines homoscedasticity in regression analysis?

Which statement defines homoscedasticity in regression analysis?

• A. For every value of x, variances on y should be equal.
• B. For every value of x, the mean of y is constant.
• C. The residuals are perfectly correlated with x.
• D. The residuals have a non-zero mean.

Answer: (A)

In regression, the concept being tested is how the spread of the residuals behaves across the range of the predictor. Homoscedasticity means the residuals (the differences between observed and predicted values) have constant variance no matter what value x takes. In other words, the dispersion of y around the regression line is the same across all levels of x. This uniform spread is important because it underpins the accuracy of standard errors, confidence intervals, and hypothesis tests for the model’s coefficients. The statement that best reflects this is the one that says the variances on y are equal for every value of x, capturing the idea of a constant residual variance across the range of x. If the residual spread increases or decreases with x, that’s heteroscedasticity, which can lead to unreliable inferences. The other options describe different issues: a constant mean of y would imply no systematic change in predicted value with x, not about the variability of the residuals; residuals being perfectly correlated with x would indicate a near-perfect fit with little to no error, which isn’t about consistent variance; residuals having a non-zero mean suggests bias in the errors, not the issue of variance across x.

In regression, the concept being tested is how the spread of the residuals behaves across the range of the predictor. Homoscedasticity means the residuals (the differences between observed and predicted values) have constant variance no matter what value x takes. In other words, the dispersion of y around the regression line is the same across all levels of x. This uniform spread is important because it underpins the accuracy of standard errors, confidence intervals, and hypothesis tests for the model’s coefficients.

The statement that best reflects this is the one that says the variances on y are equal for every value of x, capturing the idea of a constant residual variance across the range of x. If the residual spread increases or decreases with x, that’s heteroscedasticity, which can lead to unreliable inferences.

The other options describe different issues: a constant mean of y would imply no systematic change in predicted value with x, not about the variability of the residuals; residuals being perfectly correlated with x would indicate a near-perfect fit with little to no error, which isn’t about consistent variance; residuals having a non-zero mean suggests bias in the errors, not the issue of variance across x.