Distributional conformal prediction

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 many 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 arbitrary heteroskedasticity. We establish the conditional validity under consistent estimation and also provide theoretical performance guarantees under model misspecification, overfitting, and with time series data.