PROJECTED PRINCIPAL COMPONENT ANALYSIS IN FACTOR MODELS

By Jianqing Fan, Yuan Liao, Weichen Wang. 2014

201 citations recorded by the source corpus.

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.

Subfields: Food Science, Management Science and Operations Research, Statistics and Probability

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Citations in the corpus

Sufficient Forecasting Using Factor Models (2015, methodology): “ matrix onto the sieve space. The projected PCA is to run PCA on the projected matrix Xbto extract the latent factor Fb. The advantage of this is that Fbis estimated far more accurately, according to Fan et al. (2015a). Note that the projection matrix P was constructed by using the additive spline m”
Large Dimensional Latent Factor Modeling with Missing Observations and Applications to Causal Inference (2019, unknown): “2 When a panel has missing entries, a common approach is to estimate the factor (2)A current active research topic is to extend the constant loading factor model to a time-varying loading model by using a projection in the cross-section dimension (Fan et al., 2016; Kelly et al., 2018), a local windo”
Factors That Fit the Time Series and Cross-Section of Stock Returns (2018, unknown): “(2017), Fan et al. (2016), and Lettau and Pelger (2018b).”
An Overview on the Estimation of Large Covariance and Precision Matrices (2015, unknown): “Before introducing the estimator of Σ in the case of latent factors, we first elucidate why PCA can be used for the factor analysis when the number of variables is large.”
Forest Through the Trees: Building Cross-Sections of Stock Returns (2020, background): “Kelly, Pruitt, and Su (2019) and Fan, Liao, and Wang (2016) explicitly model stock loadings on latent factors as a function of characteristics, and as a result apply PCA to managed portfolios that represent linear (and nonlinear) projections of asset returns on characteristics.”
Understanding Systematic Risk: A High‐Frequency Approach (2020, unknown): “Kelly, Pruitt, and Su (2017) and Fan, Liao, and Wang (2016b) apply a version of PCA to projected portfolio data.”
Shrinking the Cross Section∗ (2017, unknown): “Fan et al. (2016) and Kelly et al. (2017) extend the standard PCA approach to allow for time-varying factor loading and apply it to extract common factors from the cross-section of individual stocks. Kelly et al. (2017) and Kozak et al. (2017) further show that an SDF constructed using few such domi”
Enhancing Interpretability in Factor Analysis by Means of Mathematical Optimization (2019, methodology): “The methodologies proposed in Fan et al. (2016); Li and Jung (2017); Taeb et al. (2017); Taeb and Chandrasekaran (2018) fit a model in which the explanatory variables are integrated together with the factors.”
State-Varying Factor Models of Large Dimensions (2018, background): “Ross (1976), Chamberlain and Rothschild (1983) and Ludvigson and Ng (2009), and in macro-economics, e.”