Projected Principal Component Analysis in Factor Models

This paper introduces a Projected Principal Component Analysis (Projected-PCA), which is based on the projection of the data matrix onto a given linear space before performing the principal component analysis. When it applies to high-dimensional factor analysis, the projection removes idiosyncratic noisy 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 the factor loading matrices are related to the projected linear space, 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. The covariates effect 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. Finally, the proposed method is illustrated by both simulated data and the returns of the components of the S&P 500 index