We consider the problem of constructing distribution-free prediction intervals for multi-step time series forecasting, with a focus on the temporal dependencies inherent in multi-step forecast errors. We establish that the optimal $h$-step-ahead forecast errors exhibit serial correlation up to lag $(h-1)$ under a general non-stationary autoregressive data generating process. To leverage these properties, we propose the Autocorrelated Multi-step Conformal Prediction (AcMCP) method, which effectively incorporates autocorrelations in multi-step forecast errors, resulting in more statistically efficient prediction intervals. This method guarantees asymptotic marginal coverage for multi-step prediction intervals, though we note that, for finite samples, the coverage error admits an upper bound that increases with the forecasting horizon. Additionally, we extend several easy-to-implement conformal prediction methods, originally designed for single-step forecasting, to accommodate multi-step scenarios. Through empirical evaluations, including simulations and applications to data, we demonstrate that AcMCP achieves coverage that closely aligns with the target within local windows, while providing adaptive prediction intervals that effectively respond to varying conditions.

翻译：本文研究多步时间序列预测中无需分布假设的预测区间构建问题，重点关注多步预测误差中固有的时间依赖性。我们证明，在一般的非平稳自回归数据生成过程下，最优的$h$步超前预测误差在滞后$(h-1)$阶内存在序列相关性。为利用这一特性，我们提出了自相关多步共形预测方法，该方法有效整合了多步预测误差中的自相关性，从而获得统计效率更高的预测区间。该方法能保证多步预测区间的渐近边际覆盖概率，但需注意在有限样本下，其覆盖误差存在随预测步长增加的上界。此外，我们将几种原本为单步预测设计的易实施共形预测方法扩展至多步预测场景。通过模拟实验和实际数据应用的实证评估，我们证明自相关多步共形预测方法能在局部窗口内实现与目标值高度吻合的覆盖概率，同时提供能有效适应不同条件的自适应预测区间。