Principal Components and Regularized Estimation of Factor Models

Abstract

It is known that the common factors in a large panel of data can be consistently estimated by the method of principal components, and principal components can be constructed by iterative least squares regressions. Replacing least squares with ridge regressions turns out to have the effect of shrinking the singular values of the common component and possibly reducing its rank. The method is used in the machine learning literature to recover low-rank matrices. We study the procedure from the perspective of estimating a minimum-rank approximate factor model. We show that the constrained factor estimates are biased but can be more efficient in terms of mean-squared errors. Rank consideration suggests a data-dependent penalty for selecting the number of factors. The new criterion is more conservative in cases when the nominal number of factors is inflated by the presence of weak factors or large measurement noise. The framework is extended to incorporate a priori linear constraints on the loadings. We provide asymptotic results that can be used to test economic hypotheses.

References in corpus (5)

• The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices
• Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
• Sparse principal component analysis and iterative thresholding
• Stable Principal Component Pursuit
• Certifiably Optimal Low Rank Factor Analysis

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• Feasible Generalized Least Squares for Panel Data with Cross-sectional and Serial Correlations
• Machine Learning Methods Economists Should Know About
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• Inference for Heterogeneous Effects using Low-Rank Estimation of Factor Slopes
• Square-root nuclear norm penalized estimator for panel data models with approximately low-rank unobserved heterogeneity