Interpretable Sparse Proximate Factors for Large Dimensions

Abstract

This article proposes sparse and easy-to-interpret proximate factors to approximate statistical latent factors. Latent factors in a large-dimensional factor model can be estimated by principal component analysis (PCA), but are usually hard to interpret. We obtain proximate factors that are easier to interpret by shrinking the PCA factor weights and setting them to zero except for the largest absolute ones. We show that proximate factors constructed with only 5%–10% of the data are usually sufficient to almost perfectly replicate the population and PCA factors without actually assuming a sparse structure in the weights or loadings. Using extreme value theory we explain why sparse proximate factors can be substitutes for non-sparse PCA factors. We derive analytical asymptotic bounds for the correlation of appropriately rotated proximate factors with the population factors. These bounds provide guidance on how to construct the proximate factors. In simulations and empirical analyses of financial portfolio and macroeconomic data, we illustrate that sparse proximate factors are close substitutes for PCA factors with average correlations of around 97.5%, while being interpretable.