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 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.

Key words and phrases: Distributional Data, Model Checking, Optimal Transport, Random Object, Wasserstein Distance 1

Information

Preprint No.SS-2026-0333
Manuscript IDSS-2026-0333
Complete AuthorsChenxiao Dai, Feiyu Jiang, Dong Li
Corresponding AuthorsFeiyu Jiang
Emailsjiangfy@fudan.edu.cn

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Acknowledgments

We would like to thank Prof. Xiaofeng Shao for his contributions to an earlier version of

this work. We are also grateful to the editor, associate editor, and two referees for their

insightful comments and suggestions, which have substantially improved the presentation

and quality of this paper. Jiang is supported in part by the National Key R&D Program of

China (No. 2024YFA1015700), and by the NSFC (Nos. 72522009, 12331009 and 72432002).

Li is supported by the NSFC (No.72471127).