Testing for the Martingale Difference Hypothesis in Multivariate Time Series Models

Guochang Wang, Ke Zhu, Xiaofeng Shao

Research output: Contribution to journalArticlepeer-review


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 (Formula presented.), 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.

Original languageEnglish (US)
Pages (from-to)980-994
Number of pages15
JournalJournal of Business and Economic Statistics
Issue number3
StatePublished - 2022


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty


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