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We propose a generalization of the linear panel quantile regression model toaccommodate both \textit{sparse} and \textit{dense} parts: sparse means whilethe number of covariates available is large, potentially only a much smallernumber of them have a nonzero impact on each conditional quantile of theresponse variable; while the dense part is represent by a low-rank matrix thatcan be approximated by latent factors and their loadings. Such a structureposes problems for traditional sparse estimators, such as the$\ell_1$-penalised Quantile Regression, and for traditional latent factorestimator, such as PCA. We propose a new estimation procedure, based on theADMM algorithm, consists of combining the quantile loss function with $\ell_1$\textit{and} nuclear norm regularization. We show, under general conditions,that our estimator can consistently estimate both the nonzero coefficients ofthe covariates and the latent low-rank matrix.  Our proposed model has a "Characteristics + Latent Factors" Asset PricingModel interpretation: we apply our model and estimator with a large-dimensionalpanel of financial data and find that (i) characteristics have sparserpredictive power once latent factors were controlled (ii) the factors andcoefficients at upper and lower quantiles are different from the median.

High Dimensional Latent Panel Quantile Regression with an Application to Asset Pricing