Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

Uncertainty quantification for time series nowcasting and forecasting is crucial in many areas such as climate science, epidemiology, industrial engineering, and macroeconomics. Ideally, the forecaster would generate a prediction interval at each time period that is calibrated in the sense that the fraction of intervals covering the true outcomes is approximately equal to the target coverage level in the long run. Classical approaches for generating prediction intervals are mostly model-based Box and Jenkins [1976], Engle [1982a], Stock and Watson [2010], Brown [1964], Jorda [2005]. However, time series models are often mis-specified due to nonstationarity or changing environments. As a result, the model-based prediction intervals tend to be poorly calibrated (see for instance the gray curves in Figure 1). Moreover, many forecasters have upgraded their workflows by incorporating black-box machine learning algorithms [e.g. Taylor and Letham, 2018, Makridakis et al., 2018, Herzen et al., 2022], for which valid uncertainty quantification proves to be challenging.