Wasserstein Exponential Smoothing for Distributional Time Series Forecasting

Distributional time series arise when each temporal observation is a probability distribution rather than a scalar. We propose Wasserstein exponential smoothing (WES), a one-parameter recursive forecasting method for distributional time series on ℝ. The method adapts the practical logic of classical exponential smoothing to probability distributions by updating forecast distributions along Wasserstein geodesics. This yields a simple filter that can be applied directly to empirical distributions without parametric density modeling. We estimate the smoothing parameter by minimizing an in-sample Wasserstein prediction loss and establish consistency under a distributional local-level data-generating process. In applications to high-frequency equity-index return distributions and household electricity-demand distributions, WES attains the lowest one-step-ahead Wasserstein prediction error among existing distributional autoregressive and regression-based benchmarks for all 20 series considered, and is retained in the 90% model confidence set in every case.