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Testing the martingale difference hypothesis in high dimension

Author

Listed:
  • Jinyuan Chang
  • Qing Jiang
  • Xiaofeng Shao
Abstract
In this paper, we consider testing the martingale difference hypothesis for high-dimensional time series. Our test is built on the sum of squares of the element-wise max-norm of the proposed matrix-valued nonlinear dependence measure at different lags. To conduct the inference, we approximate the null distribution of our test statistic by Gaussian approximation and provide a simulation-based approach to generate critical values. The asymptotic behavior of the test statistic under the alternative is also studied. Our approach is nonparametric as the null hypothesis only assumes the time series concerned is martingale difference without specifying any parametric forms of its conditional moments. As an advantage of Gaussian approximation, our test is robust to the cross-series dependence of unknown magnitude. To the best of our knowledge, this is the first valid test for the martingale difference hypothesis that not only allows for large dimension but also captures nonlinear serial dependence. The practical usefulness of our test is illustrated via simulation and a real data analysis. The test is implemented in a user-friendly R-function.

Suggested Citation

  • Jinyuan Chang & Qing Jiang & Xiaofeng Shao, 2022. "Testing the martingale difference hypothesis in high dimension," Papers 2209.04770, arXiv.org, revised Sep 2022.
  • Handle: RePEc:arx:papers:2209.04770
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    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis

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