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Large covariance estimation by thresholding principal orthogonal complements

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  • Jianqing Fan
  • Yuan Liao
  • Martina Mincheva

Abstract

This paper deals with estimation of high-dimensional covariance with a conditional sparsity structure, which is the composition of a low-rank matrix plus a sparse matrix. By assuming sparse error covariance matrix in a multi-factor model, we allow the presence of the 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. 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 specic 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, including the spectral norm. 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 veried by extensive simulation studies.

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Bibliographic Info

Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society: Series B (Statistical Methodology).

Volume (Year): 75 (2013)
Issue (Month): 4 (09)
Pages: 603-680

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Handle: RePEc:bla:jorssb:v:75:y:2013:i:4:p:603-680

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Cited by:
  1. Matteo Barigozzi & Christian Brownlees, 2013. "Nets: Network Estimation for Time Series," Working Papers 723, Barcelona Graduate School of Economics.
  2. Natalia Bailey & M. Hashem Pesaran & L. Vanessa Smith, 2014. "A Multiple Testing Approach to the Regularisation of Large Sample Correlation Matrices," CESifo Working Paper Series 4834, CESifo Group Munich.
  3. Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2014. "Estimation of the Global Minimum Variance Portfolio in High Dimensions," Papers 1406.0437, arXiv.org.
  4. Fan, Jianqing & Liao, Yuan & Shi, Xiaofeng, 2013. "Risks of large portfolios," MPRA Paper 44206, University Library of Munich, Germany.
  5. Bai, Jushan & Liao, Yuan, 2012. "Efficient Estimation of Approximate Factor Models," MPRA Paper 41558, University Library of Munich, Germany.
  6. Taras Bodnar & Arjun K. Gupta & Nestor Parolya, 2013. "On the Strong Convergence of the Optimal Linear Shrinkage Estimator for Large Dimensional Covariance Matrix," Papers 1308.2608, arXiv.org, revised Jun 2014.

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