Inference for prediction errors is critical in time series forecasting pipelines. However, providing statistically meaningful uncertainty intervals for prediction errors remains relatively under-explored. Practitioners often resort to forward cross-validation (FCV) for obtaining point estimators and constructing confidence intervals based on the Central Limit Theorem (CLT). The naive version assumes independence, a condition that is usually invalid due to time correlation. These approaches lack statistical interpretations and theoretical justifications even under stationarity. This paper systematically investigates uncertainty intervals for prediction errors in time series forecasting. We first distinguish two key inferential targets: the stochastic test error over near future data points, and the expected test error as the expectation of the former. The stochastic test error is often more relevant in applications needing to quantify uncertainty over individual time series instances. To construct prediction intervals for the stochastic test error, we propose the quantile-based forward cross-validation (QFCV) method. Under an ergodicity assumption, QFCV intervals have asymptotically valid coverage and are shorter than marginal empirical quantiles. In addition, we also illustrate why naive CLT-based FCV intervals fail to provide valid uncertainty intervals, even with certain corrections. For non-stationary time series, we further provide rolling intervals by combining QFCV with adaptive conformal prediction to give time-average coverage guarantees. Overall, we advocate the use of QFCV procedures and demonstrate their coverage and efficiency through simulations and real data examples.

翻译：预测误差的推断在时间序列预测流程中至关重要。然而，为预测误差提供具有统计意义的不确定性区间这一课题仍相对缺乏探索。实践者通常采用前向交叉验证（FCV）获取点估计量，并基于中心极限定理（CLT）构建置信区间。朴素版本假设数据独立，但该条件因时间相关性而通常不成立。即便在平稳性假设下，这些方法也缺乏统计解释与理论依据。本文系统研究时间序列预测中预测误差的不确定性区间。我们首先区分两类关键推断目标：面向近期未来数据点的随机测试误差，以及作为前者期望值的期望测试误差。在需要量化个体时间序列实例不确定性的应用中，随机测试误差往往更具相关性。为构建随机测试误差的预测区间，我们提出基于分位数的前向交叉验证（QFCV）方法。在遍历性假设下，QFCV区间具有渐近有效的覆盖概率，且比边际经验分位数更短。此外，我们还阐释了为何基于CLT的朴素FCV区间即便经过特定修正，也无法提供有效的不确定性区间。针对非平稳时间序列，我们进一步将QFCV与适应性共形预测结合，提出滚动区间以提供时间平均覆盖保证。总体而言，我们倡导使用QFCV方法，并通过模拟实验与真实数据示例证明其覆盖效率与有效性。