Distributional conformal prediction

-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.