Diagnostic Checking for Wasserstein Autoregression

TL;DR

This paper introduces diagnostic tests for Wasserstein autoregressive models, enabling effective assessment of model adequacy in distributional time series, with proven theoretical properties and demonstrated practical effectiveness.

Contribution

It develops novel portmanteau-type diagnostic tests for Wasserstein autoregression, including error autocorrelation functions and sample-splitting methods, with proven asymptotic distributions.

Findings

Tests effectively detect model mis-specification

Asymptotic chi-square distribution established

Simulation and empirical results confirm reliability

Abstract

Wasserstein autoregression provides a robust framework for modeling serial dependence among probability distributions, with wide-ranging applications in economics, finance, and climate science. In this paper, we develop portmanteau-type diagnostic tests for assessing the adequacy of Wasserstein autoregressive models. By defining autocorrelation functions for model errors and residuals in the Wasserstein space, we construct two related tests: one analogous to the classical McLeod type test, and the other based on the sample-splitting approach of Davis and Fernandes(2025). We establish that, under mild regularity conditions, the corresponding test statistics converge in distribution to chi-square limits. Simulation studies and empirical applications demonstrate that the proposed tests effectively detect model mis-specification, offering a principled and reliable diagnostic tool for distributional time series analysis.