• 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 complEtement 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 a specific example. 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 verified by extensive simulation studies.

• Paper: pdf file

• Manuscript: pdf file
  This is the first version, where the number of factors is assumed to be known, allowed to increase.

• Slides: pdf file

• Matlab functions: POET , Cmin

• Related literature:

  Fan, Liao and Mincheva (2012)
  High dimensional covariance matrix estimation in approximate factor models.
  Ann. Statist. 39, 3320-3356.

  Fan, Fan and Lv (2008)
  High dimensional covariance matrix estimation using a factor model.
  Journal of Econometrics. 147, 186-197.

  Cai and Liu (2011)
  Adaptive thresholding for sparse covariance matrix estimation.
  JASA.106, 672-684.

• Presentations:

  2012

  • Midwest Econometrics Group, Kentucky

  2011

  • Cornell University, Joint Econometrics and Statistics
  • Northwestern University, Department of Statistics