Summary: We propose a generalization of the linear panel quantile regression model to accommodate both sparse and dense parts: sparse means that while the number of covariates available is large, potentially only a much smaller number of them have a nonzero impact on each conditional quantile of the response variable; while the dense part is represent by a low-rank matrix that can be approximated by latent factors and their loadings. Such a structure poses problems for traditional sparse estimators, such as the \({\ell_1}\)-penalized quantile regression, and for traditional latent factor estimators such as PCA. We propose a new estimation procedure, based on the ADMM algorithm, that consists of combining the quantile loss function with \({\ell_1}\) and nuclear norm regularization. We show, under general conditions, that our estimator can consistently estimate both the nonzero coefficients of the covariates and the latent low-rank matrix. This is done in a challenging setting that allows for temporal dependence, heavy-tail distributions and the presence of latent factors.
Our proposed model has a “Characteristics + Latent Factors” Quantile Asset Pricing Model interpretation: we apply our model and estimator with a large-dimensional panel of financial data and find that (i) characteristics have sparser predictive power once latent factors were controlled and (ii) the factors and coefficients at upper and lower quantiles are different from the median.