  This paper develops estimation and inference methods for conditional quantile
factor models. We first introduce a simple sieve estimation, and establish
asymptotic properties of the estimators under large $N$. We then provide a
bootstrap procedure for estimating the distributions of the estimators. We also
provide two consistent estimators for the number of factors. The methods allow
us not only to estimate conditional factor structures of distributions of asset
returns utilizing characteristics, but also to conduct robust inference in
conditional factor models, which enables us to analyze the cross section of
asset returns with heavy tails. We apply the methods to analyze the cross
section of individual US stock returns.
