Residuals Math Example 4
Follow the full solution, then compare it with the other examples linked below.
Example 4
hardA residual plot shows residuals increasing in magnitude as increases (fan shape). What assumption is violated, and what does this mean for the validity of hypothesis tests in regression?
Solution
- 1 Violated assumption: homoscedasticity (constant variance of residuals) — instead, we have heteroscedasticity (variance increases with fitted values)
- 2 Impact on hypothesis tests: standard errors of regression coefficients are incorrectly calculated; t-tests for slopes and confidence intervals are invalid
- 3 Solution: transform y (log transform often works for fan patterns); use heteroscedasticity-robust standard errors (HC standard errors)
Answer
Fan-shaped residuals violate homoscedasticity. Standard errors and hypothesis tests are invalid; use transformation or robust SEs.
Regression assumptions must hold for inference to be valid. Heteroscedasticity is particularly serious because it invalidates all standard errors while leaving point estimates (slopes) unaffected. Residual plots are essential diagnostic tools that alone cannot reveal.
About Residuals
The difference between an observed value and its predicted value from a regression model: (observed minus predicted).
Learn more about Residuals →More Residuals Examples
Example 1 easy
Given [formula], and observed point [formula]: calculate the residual and interpret whether the mode
Example 2 mediumFive observed and predicted values: [formula]: [formula]. Calculate all residuals, verify they sum t
Example 3 easyA regression model gives residuals: [formula]. Are these residuals consistent with a valid LSRL? Cal