Testing for the Martingale Difference Hypothesis in Multivariate Time Series Models

Abstract

This article proposes a general class of tests to examine whether the error term is a martingale difference sequence in a multivariate time series model with parametric conditional mean. These new tests are formed based on recently developed martingale difference divergence matrix (MDDM), and they provide formal tools to test the multivariate martingale difference hypothesis in the literature for the first time. Under suitable conditions, the asymptotic null distributions of these MDDM-based tests are established. Moreover, these MDDM-based tests are consistent to detect a broad class of fixed alternatives, and have nontrivial power against local alternatives of order $n^{-1/2}$, where n is the sample size. Since the asymptotic null distributions depend on the data generating process and the parameter estimation, a wild bootstrap procedure is further proposed to approximate the critical values of these MDDM-based tests, and its theoretical validity is justified. Finally, the usefulness of these MDDM-based tests is illustrated by simulation studies and one real data example.

Keywords:

• Martingale difference divergence matrix
• Martingale difference hypothesis
• Multivariate time series models
• Specification test
• Wild bootstrap

Supplementary Materials

The supplementary materials contain some additional simulation results and the Appendix of this article.

Acknowledgments

The authors greatly appreciate the very helpful comments and suggestions of two anonymous referees, associate editor, and co-editor.

Additional information

Funding

Wang’s research is partially supported by National Social Science Fund of China (no. 20BTJ041), Guangdong Basic and Applied Basic Research Foundation (no. 2020A1515010821), and Fundamental Research Funds for the Central Universities (no. 12619624). Zhu’s research is partially supported by GRF, RGC of Hong Kong (nos. 17306818 and 17305619), NSFC (nos. 11690014 and 11731015), Seed Fund for Basic Research (no. 201811159049), and Fundamental Research Funds for the Central Universities (no. 19JNYH08). Shao’s research is partially supported by NSF-DMS (nos. 1807023 and 2014018).