高维白噪声的测试
On testing for high-dimensional white noise
摘要：
白噪声检验是统计学中的一个经典而重要的问题，特别是对于时间序列建模和线性回归的诊断检验。对于高维时间序列，维度$p$相对于样本大小$T$而言较大，流行的综合检验包括多元Hosking和Li-McLeod检验都是非常保守的，导致显著的功率损失。为了开发更相关的测试高维情况下，我们提出了一个portmanteau型的检验统计量，这是第一个$q$滞后样本自协方差矩阵的平方奇异值之和。因此，它封装了所有组件系列内和跨所有组件系列的所有系列相关性（直到时滞$q$）。利用随机矩阵理论的工具，并假设$p$和$T$都发散到无穷大，我们推导出在零和特定VMA（1）备择假设下检验统计量的渐近正态性。由于检验的实际实施需要了解总体横截面协方差阵的三个特征常数和标准化新息的四阶矩的值，因此提出了这些参数的非平凡估计，并将其整合为实际可用的检验。广泛的模拟证实了新的测试具有精确的大小和令人满意的功率为大范围的有限$(p,T)$组合的有限样本性能优良，从而确保在实践中的广泛适用性。特别是，新的测试始终优于传统的Hosking和Li-McLeod测试。
英文摘要：
Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension $p$ is large in relation to the sample size $T$, the popular omnibus tests including the multivariate Hosking and Li-McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first $q$ lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (upto the time lag $q$) within and across all component series. Using the tools from random matrix theory and assuming both $p$ and $T$ diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, non trivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite $(p,T)$ combinations, therefore ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li-McLeod tests.
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