Probabilistic time series forecasting has attracted increasing attention in financial applications due to the need to quantify risk and uncertainty in future observations. We propose ProbRes, a post-hoc probabilistic calibration method that explicitly learns and incorporates volatility dynamics into probabilistic forecasting, enabling effective handling of heteroskedastic data. During training, ProbRes employs two architecture-agnostic modules to separately model the conditional mean and conditional volatility. At the inference stage, it generates predictive distributions by resampling normalized residuals. ProbRes is applicable to both univariate and multivariate time series and remains robust under a wide range of error distributions, including non-Gaussian innovations with conditional heteroskedasticity. Theoretical results demonstrate ProbRes's validity and experiments on both synthetic and real-world datasets show that ProbRes accurately captures predictive distributions and produces well-calibrated prediction intervals.

翻译：概率时间序列预测因其在量化未来观测值中的风险与不确定性方面的需求，在金融应用中日益受到关注。本文提出ProbRes，一种事后概率校准方法，该方法通过显式学习并将波动率动态融入概率预测，从而有效处理异方差数据。在训练阶段，ProbRes采用两个与架构无关的模块分别建模条件均值与条件波动率。在推理阶段，它通过重采样标准化残差生成预测分布。ProbRes适用于单变量与多变量时间序列，并在包括含条件异方差的非高斯新息在内的广泛误差分布下保持稳健性。理论结果证明了ProbRes的有效性，在合成及真实数据集上的实验表明，ProbRes能够准确捕捉预测分布并生成校准良好的预测区间。