Bayesian Hierarchical Quantile Factor Model and Its Application

Factor analysis aims to utilize the correlations between variables to extract common factors, which are used to verify the quantitative relationship between manifest variables and latent variables. Traditional factor models focus on inferring the mean structure of the data. However, in certain specific scenarios, research not only involves the impact of latent factors on the mean but also the influence of the entire response distribution represented by quantiles. Therefore, this paper combines quantile regression with factor models, utilizing a mixture of generalized asymmetric Laplace distributions, to construct a hierarchical quantile factor model (denoted as (QFM_GAL)) within a Bayesian framework, and presents an MCMC algorithm based on Metropolis-Hastings sampling. Simulation and case studies demonstrate that the Bayesian hierarchical quantile factor analysis method exhibits robustness against outliers and extreme quantiles. Simultaneously, this method allows factor loadings and common factors to vary with quantiles, thereby providing these common factors with more realistic theoretical interpretations in practical applications.