  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.
