多维和高维白噪声检验：理论、方法及应用

白噪声检验是各类时间序列模型设定检验的核心问题，但多维与高维白噪声检验仍然面临诸多挑战。本项目研究多维及高维白噪声检验的理论、方法及应用。在理论与方法创新层面，我们首先基于时域中自相关函数显著性的合成检验，探讨检验统计量在未知相依结构和估计效应双重影响下的渐近分布，并利用随机加权自助法得到检验量的临界值，推导自助法的有效性。其次，为克服时域下检验量的非一致性，我们基于谱密度的周期图估计，构造一致的Cramér-von Mises型检验，并利用分块随机加权自助法得到检验量的临界值，推导其大样本性质。最后，我们将基于时域和频域的多维白噪声检验方法扩展到高维情形。当数据维度随样本量增加而趋于无穷时，我们借助随机矩阵理论，研究相应检验量的渐近行为。在应用层面，利用我们提出的新的白噪声检验方法，重新检视中国金融市场高维资产组合的相关性，为高维投资组合提供可靠的建模程序。

White noise checking is essential in time series model specification tests. However, it’s of many challenges in multivariate and high-dimensional white noise checking. In this project, we investigate the theory, methodology, and applications of white noise checking in multivariate and high-dimensional situations. First, in the time domain, based on the Portmanteau tests of the joint significances of autocorrelation functions, we discuss the asymptotic distribution of test statistics involving both of the unknown dependent structure of errors and estimation effects. Then, we employ the random weighting bootstrap to obtain the critical values of the test statistics and justify its validity. Second, in order to overcome the inherent inconsistency of the Portmanteau tests, we construct a consistent Cramér-von Mises type test based on the periodogram of the spectral density function, and propose the block random weighting bootstrap to obtain the critical value of the test statistics and derive its asymptotic validity. Finally, we extend the multivariate white noise checking procedures in both time and frequency domains to the high-dimensional framework. When the data dimension tends to be infinity as the sample size increases, we use random matrix theory to study the asymptotic behavior of the corresponding test statistics. In applications, we apply the new testing procedures we proposed in this project to re-examine the correlations of high dimensional asset portfolios in the Chinese financial market and provide reliable modeling procedures for high- dimensional investment portfolios.