Estimating Large Covariance Matrices with Covariates

• Activity Number: 334
• Type: Invited
• Date/Time: Tuesday, August 11, 2015 : 10:30 AM to 12:20 PM
• Sponsor: Business and Economic Statistics Section
• Author(s): Jianqing Fan* and Yuan Liao and Weichen Wang
• Companies: Princeton University and University of Maryland and Princeton University
• Keywords: Principal Component Analysis ; Factor Analysis ; Large covariance matrices ; Factor loadings ; specification tests

Abstract

We propose a flexible factor model for estimating large covariance matrices with covariates and introduce a Projected Principal Component Analysis (Projected-PCA) technique, which stengthens signals-to-noise ratios. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine and that they can be estimated accurately when the dimensionality is large, even when the sample size is finite. In an effort to more accurately estimating factor loadings, we propose a flexible semi-parametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. 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. 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 extensive numerical studies.