During exploratory analysis of model residuals, you create a normal Q-Q plot. The plotted points form an S-shape: observations in the left tail fall below the 45-degree reference line, the middle portion stays near the line, and observations in the right tail rise above it. Which conclusion and next step most appropriately address this diagnostic?
The residuals are right-skewed; apply a logarithmic transformation to reduce skewness before refitting the model.
The residuals have lighter tails than a normal distribution; a uniform or other thin-tailed distribution will suffice without further changes.
The residuals are left-skewed; square the residuals to symmetrize the distribution before re-estimating the model.
The residuals exhibit heavy tails relative to a normal distribution; refit the model with a heavy-tailed or robust error distribution (e.g., Student-t).
The described S-shape-left tail below the line and right tail above-indicates the sample distribution has heavier (long) tails than a normal distribution, i.e., it is leptokurtic. Heavy-tailed residuals violate the normal-error assumption, so switching to a model or error distribution that can accommodate fat tails (such as a t-distribution, Laplace errors, or otherwise using robust estimation) is the correct follow-up. A right-skew pattern would curve consistently upward (below the line on the left and above on the right) across the entire range, not just the extremes. Light-tailed (platykurtic) data show the opposite-tails inside the line and the center above it. A left-skew pattern would curve downward (above on the left, below on the right). Therefore, only the heavy-tail interpretation paired with a robust or heavy-tailed model is appropriate.
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What does it mean for residuals to be heavy-tailed (leptokurtic)?
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