Diagnostic Checking for Wasserstein Autoregression

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 the first-order autoregressive transport map (ATM(1)) model.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 and an additional cancellation condition for the splitting test statistic, 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.