Open Access
February 2016 Projected principal component analysis in factor models
Jianqing Fan, Yuan Liao, Weichen Wang
Ann. Statist. 44(1): 219-254 (February 2016). DOI: 10.1214/15-AOS1364


This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employs principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semiparametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates’ effects on the factor loadings are further modeled by the additive model via sieve approximations. By using the newly proposed Projected-PCA, the rates of convergence of the smooth factor loading matrices are obtained, which are much faster than those of the conventional factor analysis. The convergence is achieved even when the sample size is finite and is particularly appealing in the high-dimension-low-sample-size situation. This leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings. The proposed method is illustrated by both simulated data and the returns of the components of the S&P 500 index.


Download Citation

Jianqing Fan. Yuan Liao. Weichen Wang. "Projected principal component analysis in factor models." Ann. Statist. 44 (1) 219 - 254, February 2016.


Received: 1 January 2015; Revised: 1 July 2015; Published: February 2016
First available in Project Euclid: 10 December 2015

zbMATH: 1331.62295
MathSciNet: MR3449767
Digital Object Identifier: 10.1214/15-AOS1364

Primary: 62H25
Secondary: 62H15

Keywords: high-dimensionality , loading matrix modeling , Semiparametric factor models , sieve approximation

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 1 • February 2016
Back to Top