Online conformal inference for multi-step time series forecasting

TL;DR

The paper tackles distribution-free uncertainty quantification for multi-step time series forecasts under nonstationary dynamics. It develops AcMCP, an online conformal prediction framework that explicitly models autocorrelation in multi-step forecast errors, and extends existing conformal tools (MSCP, MWCP, MACP, MPID) to horizons beyond one step. The authors establish that optimal $h$-step-ahead forecast errors follow an approximate MA$(h-1)$ structure and prove that AcMCP achieves asymptotic marginal coverage with finite-sample bounds that grow with horizon. Empirical results on simulated and real datasets show AcMCP delivering coverage close to target within local windows while providing adaptive, informative prediction intervals, and the authors provide an open-source R package for implementation. The work offers a practical, distribution-free approach to reliable multi-step uncertainty quantification in time series with potential broad impact on forecasting in economics, energy, and related domains.

Abstract

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.

Figures (11)

• Figure 1: Online learning framework with sequential splits. White: unused data; Gray: training data; Black: forecasts in calibration set; Blue: forecasts in test set.
• Figure 2: AR(2) simulation results showing rolling coverage, mean and median interval width for each forecast horizon. The displayed curves are smoothed over a rolling window of size $500$. The target coverage level is $1-\alpha=0.9$.
• Figure 3: AR(2) simulation results showing boxplots of the rolling coverage and interval width for each method across different forecast horizons. The red dashed lines show the target coverage level, while the blue dashed lines indicate the median interval width of the AcMCP method.
• Figure 4: AR(2) simulation results showing the prediction interval bounds for the MACP, MPI, and AcMCP methods over a truncated period of length 500.
• Figure 5: Nonlinear simulation results showing rolling coverage, mean and median interval width for each forecast horizon. The displayed curves are smoothed over a rolling window of size $100$. The target coverage level is $1-\alpha=0.9$.

Theorems & Definitions (15)

• Proposition 1: Autocorrelations of multi-step optimal forecast errors
• proof
• Proposition 2: MA$(h-1)$ process for $h$-step-ahead optimal forecast errors
• proof
• Proposition 3
• proof
• Corollary 1
• proof
• Corollary 2
• proof