期刊
JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
卷 75, 期 4, 页码 603-680出版社
OXFORD UNIV PRESS
DOI: 10.1111/rssb.12016
关键词
Approximate factor model; Cross-sectional correlation; Diverging eigenvalues; High dimensionality; Low rank matrix; Principal components; Sparse matrix; Thresholding; Unknown factors
资金
- National Institutes of Health [R01GM100474-01, R01-GM072611]
- Bendheim Center for Finance at Princeton University
- [DMS-0704337]
- EPSRC [EP/J017213/1] Funding Source: UKRI
The paper deals with the estimation of a high dimensional covariance with a conditional sparsity structure and fast diverging eigenvalues. By assuming a 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 method POET' to explore such an approximate factor structure with sparsity. The POET-estimator includes the sample covariance matrix, the factor-based covariance matrix, the thresholding estimator and the adaptive thresholding estimator 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 effect 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.
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