A Simple Three-Parameter Residual Scaling Rule for Cross-Domain Probabilistic Forecasting

Practitioners need simple, robust prediction-interval methods that work across diverse domains without custom tuning. This preprint presents Empirical Residual Scaling (ERS), a three-parameter quantile rule based on level-based residuals with linear horizon scaling, and evaluates it against split conformal prediction across 11 forecasting domains spanning epidemiology, technology adoption, energy, finance, and retail.

Across 25,593 forecast instances, ERS achieves 94.9% empirical coverage at 90% nominal (95% CI: 94.7–95.2%), while split conformal achieves 95.6% (95% CI: 95.4–95.9%). The difference is statistically significant (p < 0.001) but practically small. Conformal produces meaningfully sharper intervals, with 32% lower Weighted Interval Score (p < 0.001).

A key finding is that good interval coverage does not imply good distributional calibration. ERS exhibits calibration asymmetry, under-covering lower quantiles and over-covering upper quantiles, while conformal tracks nominal quantiles more closely in the middle of the distribution, notably at the median. We document clear failure modes, including ERS under-coverage on zero-inflated data (M5 Retail: 82.4%) and conformal under-coverage on short annual series (Bass Tech: 72.4%). For applications requiring reliable quantile estimates beyond coverage, we recommend conformal approaches.