On the Inference about the Spectral Distribution of High-Dimensional Covariance Matrix Based on High-Frequency Noisy Observations

52 Pages Posted: 14 Apr 2016 Last revised: 11 Mar 2017

Ningning Xia

Shanghai University of Finance and Economics - School of Statistics and Management

Xinghua Zheng

Hong Kong University of Science & Technology (HKUST) - Department of Information Systems, Business Statistics and Operations Management

Date Written: March 1, 2017

Abstract

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-noise-type 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\v{c}enko-Pastur equation. The performance of the two approaches is examined via simulation studies under both synchronous and asynchronous observation settings.

Keywords: High-dimension, high-frequency, integrated covariance matrices, Marcenko-Pastur equation, microstructure noise

Suggested Citation

Xia, Ningning and Zheng, Xinghua, On the Inference about the Spectral Distribution of High-Dimensional Covariance Matrix Based on High-Frequency Noisy Observations (March 1, 2017). Available at SSRN: https://ssrn.com/abstract=2764654 or http://dx.doi.org/10.2139/ssrn.2764654

Ningning Xia (Contact Author)

Shanghai University of Finance and Economics - School of Statistics and Management ( email )

777 Guoding Road
Shanghai, Shanghai 200433
China

Xinghua Zheng

Hong Kong University of Science & Technology (HKUST) - Department of Information Systems, Business Statistics and Operations Management ( email )

Clear Water Bay
Kowloon
Hong Kong

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