  This paper studies the estimation of characteristic-based quantile factor
models where the factor loadings are unknown functions of observed individual
characteristics while the idiosyncratic error terms are subject to conditional
quantile restrictions. We propose a three-stage estimation procedure that is
easily implementable in practice and has nice properties. The convergence
rates, the limiting distributions of the estimated factors and loading
functions, and a consistent selection criterion for the number of factors at
each quantile are derived under general conditions. The proposed estimation
methodology is shown to work satisfactorily when: (i) the idiosyncratic errors
have heavy tails, (ii) the time dimension of the panel dataset is not large,
and (iii) the number of factors exceeds the number of characteristics. Finite
sample simulations and an empirical application aimed at estimating the loading
functions of the daily returns of a large panel of S\&P500 index securities
help illustrate these properties.
