Model Fit (Intuition) Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

A model has

R^2 = 0.99

in-sample but shows a fanning residual plot (residuals grow larger as

\hat{y}

increases). What problem does this reveal, and what are its consequences?

1
Fanning residuals = heteroscedasticity: variance of errors increases with predicted values
2
This violates regression assumption of constant variance (homoscedasticity)
3
Consequence 1: standard errors are incorrect, making hypothesis tests and confidence intervals unreliable
4
Consequence 2: predictions are less reliable for high values (where variance is largest)
5
Fix: transform y (e.g., log transform), use weighted least squares, or a robust regression method

Answer

Fanning residuals indicate heteroscedasticity — standard errors and confidence intervals are invalid despite high

R^2

A high

R^2

does not mean all regression assumptions are satisfied. Heteroscedasticity is undetectable from

R^2

alone but is visible in residual plots. It invalidates standard errors and inference, making it a serious problem despite good apparent fit.

About Model Fit (Intuition)

Model fit describes how closely a statistical model's predictions match the observed data — measured by residuals, $R^2$ , or loss functions.

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