  This paper studies large $N$ and large $T$ conditional quantile panel data
models with interactive fixed effects. We propose a nuclear norm penalized
estimator of the coefficients on the covariates and the low-rank matrix formed
by the fixed effects. The estimator solves a convex minimization problem, not
requiring pre-estimation of the (number of the) fixed effects. It also allows
the number of covariates to grow slowly with $N$ and $T$. We derive an error
bound on the estimator that holds uniformly in quantile level. The order of the
bound implies uniform consistency of the estimator and is nearly optimal for
the low-rank component. Given the error bound, we also propose a consistent
estimator of the number of fixed effects at any quantile level. To derive the
error bound, we develop new theoretical arguments under primitive assumptions
and new results on random matrices that may be of independent interest. We
demonstrate the performance of the estimator via Monte Carlo simulations.
