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.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

“…On the other hand, if the dimension is much larger than the sample size, we offset the dimensionality by assuming increased signals or sample size, without additional sparse eigenvector assumption as in sparse PCA regime. In particular, as shown in (1.1), the strong (or pervasive) factors considered in financial applications corresponds to γ j = 0 with the leading eigenvalues λ j ≍ p ; see for example Stock and Watson (2002); Bai (2003); Bai and Ng (2002); Fan, Liao and Mincheva (2013); Fan, Liao and Wang (2016). The weak or semi-strong factors considered by De Mol, Giannone and Reichlin (2008) and Onatski (2012) also imply bounded p /( n λ 1 ), with p/n bounded and λ j ≍ p θ for some θ ∈ [0, 1).…”

Supervised singular value decomposition and its asymptotic properties

“…In particular, the matrix F captures information in U that cannot be explained by Y. We note that very recently Fan et al [13] proposed a projected principal component analysis (PCA) method that generalizes the second equation of (1) to a semi-parametric model.…”

An overview of the estimation of large covariance and precision matrices

“…Additional works on semi-parametric factor models include, e.g., Park et al (2009) and Song et al (2014). Fan et al (2016b) recognized that the above semi-parametric model (7.1) might be restrictive for applications, as we do not expect that the covariates capture completely the factor loadings. They extend the model to the following more flexible semi-parametric mixed effect model:…”

“…To deal with the curse of dimensionality, we assume g k (·) to be additive: g k (X i ) = d l=1 g kl (X il ), with d = dim(X i ). Applying the projected PCA on to the semi-parametric factor model, Fan et al (2016b) showed that as p, J → ∞, T may either grow or stay constant,…”

“…Clearly under the high dimensionality, the rate of convergence is fast even if T is finite. We refer the readers to Fan et al (2016b) for more detailed discussions of the impacts of improved rates of convergence in factor models.…”