Temporal Conformal Prediction (TCP): A Distribution-Free Statistical and Machine Learning Framework for Adaptive Risk Forecasting

We propose $\mathbf{\text{Temporal}}\mathbf{\text{Conformal}}\mathbf{\text{Prediction}}\mathbf{\text{(TCP)}}$, a distribution-free framework for constructing well-calibrated prediction intervals in nonstationary time series. TCP couples a modern quantile forecaster with a rolling split-conformal calibration layer; its $\mathbf{\text{TCP-RM}}$ variant adds an online Robbins-Monro offset to steer coverage in real time. We benchmark TCP against GARCH, Historical Simulation, Quantile Regression (QR), linear QR, and Adaptive Conformal Inference (ACI) across S&P 500, Bitcoin, and Gold. Three results are consistent. First, QR baselines yield the sharpest intervals but are materially under-calibrated; even ACI remains below the 95% target. Second, TCP achieves near-nominal coverage, yielding intervals slightly wider than Historical Simulation (e.g., S&P 500: 5.21 vs.\ 5.06). Third, the RM update changes calibration only marginally at default hyperparameters. Crisis-window visualizations (March 2020) show TCP promptly expanding and contracting intervals as volatility spikes. A sensitivity study confirms robustness to hyperparameters. Overall, TCP bridges statistical inference and machine learning, providing a practical solution for calibrated risk forecasting under distribution shift.