Volatility-Corrected Prediction Intervals: Locally Adaptive Uncertainty Quantification for Probabilistic Time-Series Forecasting

Abstract

Prediction intervals from probabilistic time-series forecasters are commonly assessed by marginal coverage, the long-run proportion of observations falling inside the intervals. However, forecast errors in time series are often serially dependent and heteroskedastic, so prediction intervals can attain near-nominal marginal coverage while miscovering systematically during periods of larger local error variance. We identify and address this failure mode, which we call \emph{local scale miscalibration}. We first document the phenomenon using time-series foundation models (TSFMs): across several real-world series, their native intervals often attain near-nominal marginal coverage while exhibiting substantial temporal variation in local coverage, alternating between overcoverage and undercoverage over shorter windows. We then propose Volatility-Corrected Prediction Intervals (VCPI), a post-hoc calibration procedure that uses the base forecaster's predictive scale as a baseline, models the volatility of standardized forecast residuals with a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, and applies a multiplicative correction to produce locally adaptive prediction intervals. We justify VCPI both theoretically and empirically, comparing it against model-agnostic conformal counterparts. We provide a theoretical analysis of interval adaptation of the proposed method following abrupt changes in volatility. Empirically, we evaluate VCPI on Chronos, Moirai, and TimesFM across four real-world datasets and find that it produces more stable local coverage than conformal baselines while maintaining near-nominal marginal coverage and comparable interval width.