ON THE INFERENCE ABOUT THE SPECTRAL DISTRIBUTION OF HIGH-DIMENSIONAL COVARIANCE MATRIX BASED ON HIGH-FREQUENCY NOISY OBSERVATIONS

• 成果类型: Article
• 署名作者: Xia, Ningning; Zheng, Xinghua
• 署名单位: Shanghai University of Finance & Economics; Hong Kong University of Science & Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1558
• 发表日期: 2018
• 页码: 500-525
• 关键词: MICROSTRUCTURE NOISE; volatility; eigenvalues; models; RISK

摘要

In practice, observations are often contaminated by noise, making the resulting sample covariance matrix a signal-plus-noise sample covariance matrix. Aiming to make inferences about the spectral distribution of the population covariance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (signal) sample covariance matrices depends on that of signal-plus-noisetype sample covariance matrices. As an application, we consider inferences about the spectral distribution of integrated covolatility (ICV) matrices of high-dimensional diffusion processes based on high-frequency data with microstructure noise. The (slightly modified) pre-averaging estimator is a signal-plus-noise sample covariance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of signal sample covariance matrices and that of the population covariance matrix, enables us to propose a two-step procedure to consistently estimate the spectral distribution of ICV for a class of diffusion processes. An alternative approach is further proposed, which possesses several desirable properties: it is more robust, it eliminates the effects of microstructure noise, and the asymptotic relationship that enables consistent estimation of the spectral distribution of ICV is the standard Mar. cenko-Pastur equation. The performance of the two approaches is examined via simulation studies under both synchronous and asynchronous observation settings.