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.
翻译:在本文中, 我们考虑测试高维时间序列的 Martingale 差异假设。 我们的测试建立在拟议矩阵估值的非线性依赖度测量的元素- 最大向量的正方方和不同的时滞。 为了进行推论, 我们用高西亚近似值来估计测试统计数据的无效分布, 并提供基于模拟的方法来生成关键值。 还研究了替代数据下测试统计的无线行为。 我们的方法是非参数的。 我们的方法是非参数的, 因为空虚假设仅假设相关的时间序列是马丁ale 差异, 而没有具体说明其条件时刻的任何参数形式。 作为高斯近似法的优势, 我们的测试对未知规模的跨序列依赖性是强大的。 据我们所知, 这是对马丁格尔差异假设的第一个有效测试, 不仅允许大维度, 而且还能捕捉非线性序列依赖性。 我们的测试的实际效用通过模拟和真实的数据分析来说明。 测试是在方便用户的功能下进行。