Optimal Transport Guarantees to Nonparametric Regression for Locally Stationary Time Series

Locally stationary time series (LSTS) represent an essential modeling paradigm for capturing the nuanced dynamics inherent in time series data, whose statistical characteristics, including mean and variance, evolve smoothly over time. In this paper, we propose a conditional probability distribution estimator for LSTS through Nadaraya–Watson (NW) kernel smoothing. NW estimator leverages local kernel smoothing to approximate the conditional distribution of a response variable given its covariates. Under mild conditions, we establish optimal transport convergence guarantees to the proposed NW-based conditional probability estimator. These guarantees are initially proven in the univariate setting using the Wasserstein distance, and subsequently in a multivariate setting employing the sliced Wasserstein distance. To corroborate our theoretical findings, we conduct a wide range of numerical experiments to assess the convergence rates and showcase the practical relevance of the estimator in capturing intricate temporal dependencies in complex nonstationary phenomena.