Suppose you have a residuals plot that shows a funnel shape for the residuals. Which assumption of linear regression is being violated?

Identifying the Violation

1. Funnel Shape Explanation: A residuals plot that exhibits a funnel shape typically indicates that the variance of the residuals is not constant across all levels of the independent variable(s). This is a core aspect of what is known as homoscedasticity.

2. Assumption Being Violated: The violation corresponds to the assumption of homoscedasticity. In linear regression, one fundamental assumption is that the residuals (the differences between observed and predicted values) must exhibit constant variance (homoscedasticity) across all levels of the independent variable(s).

3. Implications of Violation: When this assumption is violated (as indicated by the funnel shape), it suggests that the variability of the errors is changing (heteroscedasticity). This can lead to inefficiencies in the regression estimates and can affect the reliability of hypothesis tests related to the regression coefficients.

In summary, a funnel-shaped residuals plot suggests that the assumption of homoscedasticity is being violated, indicating that the residuals are heteroscedastic. This variability can lead to issues in interpreting linear regression results accurately.