Testing the martingale difference hypothesis in high dimension

TL;DR

This paper introduces a novel nonparametric test for the martingale difference hypothesis in high-dimensional time series, capable of detecting nonlinear serial dependence and robust to cross-series dependence.

Contribution

It develops the first valid high-dimensional, nonlinear dependence test for martingale difference hypothesis using Gaussian approximation and simulation-based critical values.

Findings

Test effectively detects nonlinear serial dependence.

Method robust to unknown cross-series dependence.

Validated through simulations and real data analysis.

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.