On the Estimation of Integrated Covariance Matrices of High Dimensional Diffusion Processes

Annals of Statistics, Forthcoming

39 Pages Posted: 14 May 2010 Last revised: 10 Oct 2011

See all articles by Xinghua Zheng

Xinghua Zheng

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

Yingying Li

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

Date Written: October 10, 2011

Abstract

We consider the estimation of integrated covariance (ICV) matrices of high dimensional diffusion processes based on high frequency observations. We start by studying the most commonly used estimator, the realized covariance (RCV) matrix. We show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate, the limiting spectral distribution (LSD) of RCV depends on the covolatility processnot only through the targeting ICV, but also on how the covolatility process varies in time. We establish a Marcenko-Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marcenko-Pastur type theorem for RCV for a class C of diffusion processes. The results explicitly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class C, the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Marcenko-Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.

Keywords: High dimension, high frequency, integrated covariance matrix, Marcenko-Pastur equation, random matrix theory, realized covariance matrix

Suggested Citation

Zheng, Xinghua and Li, Yingying and Li, Yingying, On the Estimation of Integrated Covariance Matrices of High Dimensional Diffusion Processes (October 10, 2011). Annals of Statistics, Forthcoming, Available at SSRN: https://ssrn.com/abstract=1604271

Xinghua Zheng (Contact Author)

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

Clear Water Bay
Kowloon
Hong Kong

Yingying Li

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

Clear Water Bay
Kowloon
Hong Kong

Hong Kong University of Science & Technology (HKUST), Dept of ISOM and Dept of Finance ( email )

Clear Water Bay, Kowloon
Hong Kong

Hong Kong University of Science & Technology (HKUST) - Department of Finance ( email )

Clear Water Bay, Kowloon
Hong Kong

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