Questions: Graphical Diagnostics: Residual Plots and QQ Plots

Question 1

A residuals vs. fitted values plot shows residuals scattered tightly near zero for small fitted values, but spreading widely for large fitted values in a fan shape. What does this pattern most likely indicate?

• A. Nonlinearity — the model's functional form is incorrect and a quadratic term is needed
• B. Heteroskedasticity — error variance increases with the level of fitted values
• C. Autocorrelation — errors are correlated with prior residuals
• D. Multicollinearity — two predictors are highly correlated with each other

Answer: (B)

A fan shape — narrow on the left, wide on the right — is the classic signature of heteroskedasticity, where error variance grows with the fitted value. Nonlinearity looks different: it appears as a U-shape or arch (systematic curvature), not a spreading fan. Autocorrelation produces wave-like patterns in residuals ordered by time, not a fan. Multicollinearity affects coefficient standard errors but doesn't produce a specific residual plot pattern.

Question 2

A QQ plot of regression residuals shows points falling on the 45-degree line in the middle but curving strongly upward at the right tail and downward at the left tail, forming an S-curve. What does this indicate?

• A. The residuals are approximately normally distributed
• B. The residuals have heavy tails — extreme values are more frequent than a normal distribution predicts
• C. The residuals are right-skewed, with more large positive values than expected
• D. The model has a nonlinearity problem that should be addressed with a log transformation

Answer: (B)

An S-curve on a QQ plot — points above the line at the right tail and below at the left — indicates heavy tails (leptokurtosis). If residuals were normal, all points would track the 45-degree line. Right skew produces a different shape: points bow upward on the right only. Heavy tails matter because they affect the reliability of t-tests on individual coefficients, particularly in small samples.

Question 3

A fan-shaped pattern in a residuals vs. fitted values plot suggests that the model's functional form is wrong and a quadratic term should be added.

• T. True
• F. False

Answer: False

A fan shape indicates heteroskedasticity (non-constant error variance), not a functional form problem. The fix typically involves weighted least squares, robust standard errors, or a variance-stabilizing transformation. A U-shaped or curved pattern in the residuals indicates nonlinearity — that's when you consider adding polynomial terms or transforming variables. Confusing these two patterns leads to the wrong diagnosis and the wrong correction.

Question 4

Graphical diagnostics like residual plots are valuable partly because they show where a model violation is concentrated, information that a single test statistic cannot convey.

• T. True
• F. False

Answer: True

A Breusch-Pagan test or White test gives you a p-value: the violation is or isn't significant. A residual plot shows *which* fitted values are affected, whether the problem is smooth or driven by a few influential observations, and which variable might be driving it. This spatial information guides the correction — you can't tell from a p-value alone whether to use weighted least squares, transform a variable, or investigate outliers.

Question 5

What distinguishes a residual plot showing heteroskedasticity from one showing nonlinearity, and why does the distinction matter for how you fix the model?

Model answer: Heteroskedasticity appears as a fan shape — residuals spread out or contract systematically as fitted values increase, but with no systematic direction (positive or negative). Nonlinearity appears as a systematic curve — residuals trend positive then negative (or vice versa) across fitted values, indicating the model's mean function is misspecified. The fixes differ: heteroskedasticity calls for robust standard errors or weighted least squares; nonlinearity calls for adding nonlinear terms (quadratic, log) or transforming variables. Applying the nonlinearity fix to a heteroskedasticity problem leaves the variance problem unaddressed.

The diagnostic purpose of residual plots is to distinguish violations so you can apply the right correction. Confusing the fan pattern with the arch pattern is a common mistake. The scale-location plot (square root of absolute residuals vs. fitted values) is a cleaner tool for isolating heteroskedasticity, while the standard residuals vs. fitted plot is better for spotting nonlinearity as a curvature pattern.