Testing High-Dimensional Covariance Matrices Under the Elliptical Distribution and Beyond

30 Pages Posted: 19 Jul 2017 Last revised: 1 Feb 2018

Xinxin Yang

Hong Kong University of Science & Technology (HKUST) - School of Science

Xinghua Zheng

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

Jiaqi Chen

Harbin Institute of Technology

Li Hua

Changchun University - Department of Science

Date Written: July 13, 2017

Abstract

We study testing high-dimensional covariance matrices under a generalized elliptical model. The model accommodates several stylized facts of real data including heteroskedasticity, heavy-tailedness, asymmetry, etc. We consider the high-dimensional setting where the dimension p and the sample size n grow to infinity proportionally, and establish a central limit theorem for the linear spectral statistic of the sample covariance matrix based on self-normalized observations. The central limit theorem is different from the existing ones for the linear spectral statistic of the usual sample covariance matrix. Our tests based on the new central limit theorem neither assume a specific parametric distribution nor involve the kurtosis of data. Simulation studies show that our tests work well even when the fourth moment does not exist. Empirically, we analyze the idiosyncratic returns under the Fama-French three-factor model for S&P 500 Financials sector stocks, and our tests reject the hypothesis that the idiosyncratic returns are uncorrelated.

Keywords: covariance matrix, high-dimension, elliptical model, linear spectral statistics, central limit theorem, self-normalization

JEL Classification: C12, C55

Suggested Citation

Yang, Xinxin and Zheng, Xinghua and Chen, Jiaqi and Hua, Li, Testing High-Dimensional Covariance Matrices Under the Elliptical Distribution and Beyond (July 13, 2017). Available at SSRN: https://ssrn.com/abstract=3001811 or http://dx.doi.org/10.2139/ssrn.3001811

Xinxin Yang

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

Room 6515, 6/F, Lifts 25/26
Clear Water Bay
Kowloon
Hong Kong

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

Jiaqi Chen

Harbin Institute of Technology ( email )

92 West Dazhi Street
Nan Gang District
Harbin, 150001
China

Li Hua

Changchun University - Department of Science ( email )

6543 Satellite Road
Chaoyang District
Changchun City, Jilin Province
China

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