Large Covariance Estimation by Thresholding Principal Orthogonal Complements

57 Pages Posted: 31 Dec 2011 Last revised: 5 Jan 2013

See all articles by Jianqing Fan

Jianqing Fan

Princeton University - Bendheim Center for Finance

Yuan Liao

Rutgers, The State University of New Jersey - New Brunswick/Piscataway

Martina Mincheva

Princeton University

Date Written: December 30, 2011

Abstract

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

Keywords: High dimensionality, approximate factor model, unknown factors, principal components, sparse matrix, low-rank matrix, thresholding, cross-sectional correlation

Suggested Citation

Fan, Jianqing and Liao, Yuan and Mincheva, Martina, Large Covariance Estimation by Thresholding Principal Orthogonal Complements (December 30, 2011). Available at SSRN: https://ssrn.com/abstract=1977673 or http://dx.doi.org/10.2139/ssrn.1977673

Jianqing Fan

Princeton University - Bendheim Center for Finance ( email )

26 Prospect Avenue
Princeton, NJ 08540
United States
609-258-7924 (Phone)
609-258-8551 (Fax)

HOME PAGE: http://orfe.princeton.edu/~jqfan/

Yuan Liao (Contact Author)

Rutgers, The State University of New Jersey - New Brunswick/Piscataway ( email )

94 Rockafeller Road
New Brunswick, NJ 08901
United States

HOME PAGE: http://rci.rutgers.edu/~yl1114

Martina Mincheva

Princeton University ( email )

22 Chambers Street
Princeton, NJ 08544-0708
United States

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