High Dimensional Quantile Factor Analysis

In this paper, I develop a method that extends quantile regressions to high dimensional factor analysis. In this context, the quantile function of a panel of variables with crosssection and timeseries dimensions N and T is endowed with a factor structure. Thus, both factors and factor loadings are allowed to be quantile-specific. I provide a set of conditions under which these objects are identified, and I propose a simple two-step iterative procedure called Quantile Principal Components (QPC) to estimate them. Uniform consistency of the estimators is established under general assumptions when N,T?8 jointly. Lastly, under certain additional assumptions related to the density of the observations about the quantile of interest, and the relationship between N and T, I show that the QPC estimators are asymptotically normal with convergence rates similar to the ones derived in the traditional factor analysis literature. Monte Carlo simulations confirm the good performance of the QPC procedure, especially in non-linear environments, or when the factors affect higher moments of the observable variables and suggest that the proposed theory provides a good approximation to the finite sample distribution of the QPC estimators.