Testing Error Distribution by Kernelized Stein Discrepancy in Multivariate Time Series Models

Abstract

Knowing the error distribution is important in many multivariate time series applications. To alleviate the risk of error distribution mis-specification, testing methodologies are needed to detect whether the chosen error distribution is correct. However, the majority of existing tests only deal with the multivariate normal distribution for some special multivariate time series models, and thus cannot be used for testing the often observed heavy-tailed and skewed error distributions in applications. In this article, we construct a new consistent test for general multivariate time series models, based on the kernelized Stein discrepancy. To account for the estimation uncertainty and unobserved initial values, a bootstrap method is provided to calculate the critical values. Our new test is easy-to-implement for a large scope of multivariate error distributions, and its importance is illustrated by simulated and real data. As an extension, we also show how to test for the error distribution in copula time series models.

Keywords:

• Consistent test
• Copula time series model
• Kernelized Stein discrepancy
• Multivariate time series model
• Testing multivariate error distribution

Acknowledgments

The authors are very grateful to Feiyu Jiang, the two referees, the associate editor, and the co-editor for their constructive suggestions and comments, leading to a substantial improvement in the presentation and contents. The authors also thank Robert Mills for his professional proof-editing help.

Additional information

Funding

Zhu’s work is supported in part by the GRF, RGC of Hong Kong (nos. 17304421, 17306818, and 17305619) and the NSFC (nos. 11690014 and 11731015). Li’s work is supported in part by the NSFC (nos. 11771239 and 71973077) and the Tsinghua University Initiative Scientific Research Program, China (no. 2019Z07L01009).