Distributional conformal prediction

Prediction problems are important in many contexts. Examples include cross-sectional prediction, time series forecasting, counterfactual prediction and synthetic controls, and individual treatment effect prediction. We develop a prediction method that works in conjunction with many powerful classical methods (e.g., conventional quantile regression) as well as modern high-dimensional methods for estimating conditional distributions (e.g., quantile neural networks). Unlike many existing prediction approaches, our method is valid conditional on the observed predictors and efficient under some conditions. Importantly, our method is also robust; it exhibits unconditional coverage guarantees under model misspecification, under overfitting, and with time series data.

We propose a robust method for constructing conditionally valid prediction intervals based on models for conditional distributions such as quantile and distribution regression. Our approach can be applied to important prediction problems, including cross-sectional prediction, k–step-ahead forecasts, synthetic controls and counterfactual prediction, and individual treatment effects prediction. Our method exploits the probability integral transform and relies on permuting estimated ranks. Unlike regression residuals, ranks are independent of the predictors, allowing us to construct conditionally valid prediction intervals under heteroskedasticity. We establish approximate conditional validity under consistent estimation and provide approximate unconditional validity under model misspecification, under overfitting, and with time series data. We also propose a simple “shape” adjustment of our baseline method that yields optimal prediction intervals.