Projected Principal Component Analysis in Factor Models

Abstract

This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employees 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 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' 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.

Projected principal component analysis in factor models

“…One of the typical examples that satisfies this assumption is the semi-parametric factor model (model (1.4)). We shall study this specific type of factor model in Section 4, and prove Assumption 3.1 in the supplementary material Fan et al (2015b).…”

“…It is often satisfied when the covariance matrix $E{\mathbf{u}}_t{\mathbf{u}}_t^{\prime }$ is sufficiently sparse under the strong mixing condition. We provide primitive conditions of condition (iii) in the supplementary material Fan et al (2015b).…”

“…We expect the conclusions in the current paper to still hold if some smoothness assumptions are added for the time varying components of the covariates. Due to the space limit, we provide heuristic discussions on this case in the supplementary material of this paper Fan et al (2015b). In addition, note that in the usual factor model, Λ was assumed to be deterministic.…”

“… SUPPLEMENTARY MATERIAL Supplement: Technical proofs Fan et al (2015b) (). This supplementary material contains all the remaining proofs.…”

Large covariance estimation through elliptical factor models

“…This research area is very active, and as a result, this list of references is illustrative rather than comprehensive. To emphasize, Fan and his collaborators proposed to use factor model, which entails a conditional sparsity structure, for covariance matrix estimation (Fan, Fan and Lv, 2008; Fan, Liao and Mincheva, 2011, 2013; Fan, Liao and Wang, 2014). The model encompasses the situation of unconditional sparse covariance by setting the number of factors to zero.…”