检验高维时间序列中的鞅差假设
Testing the martingale difference hypothesis in high dimension
AI总结：
本文提出一种基于矩阵值非线性依赖度量平方和的高维鞅差假设检验方法，利用高斯近似和模拟临界值，非参数且稳健，为首个能处理大维度与非线性依赖的有效检验。
摘要：
本文考虑检验高维时间序列的鞅差假设。我们的检验基于所提出的矩阵值非线性依赖度量在不同滞后阶上的元素最大范数的平方和。为进行推断，我们通过高斯近似逼近检验统计量的零分布，并提供基于模拟的方法生成临界值。同时，研究了备择假设下检验统计量的渐近行为。我们的方法是非参数的，因为零假设仅假设所关注的时间序列是鞅差，而不指定其条件矩的任何参数形式。作为高斯近似的一个优势，我们的检验对未知幅度的跨序列依赖性具有稳健性。据我们所知，这是首个不仅允许大维度而且能捕捉非线性序列依赖的鞅差假设的有效检验。通过模拟和真实数据分析说明了我们检验的实际用途。该检验已实现为用户友好的R函数。
英文摘要：
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.
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