Robust Factor Models with Covariates
ABSTRACT: We study factor models when the latent factors can be partially explained by observed covariates. With those covariates, both the factors and loadings are identifiable up to a rotation matrix even only with a sufficiently large finite dimensions. To incorporate the explanatory power of these covariates, we propose a smoothed principal component analysis (PCA): (i) regress the data onto the observed covariates, and (ii) take the principal components of the fitted data to estimate the loadings and factors. We show that both the estimated factors and loadings can be estimated with improved rates of convergence compared to the benchmark method. The degree of improvement depends on the strength of the signals, representing the explanatory power of the covariates on the factors. The proposed estimator is robust to possibly heavy-tailed distributions, which are encountered in many high-dimensional applications for factor analysis. Empirically, our method leads to a substantial improvement on the out-of-sample forecast on the US bond excess return data.