Shape Regressions

Learning about the shape of a probability distribution, not just about its location or dispersion, is often an important goal of empirical analysis. Given a continuous random variable Y and a random vector X defined on the same probability space, the conditional distribution function (CDF) and the conditional quantile function (CQF) offer two equivalent ways of describing the shape of the conditional distribution of Y given X. To these equivalent representations correspond two alternative approaches to shape regression. One approach - distribution regression - is based on direct estimation of the conditional distribution function (CDF); the other approach - quantile regression - is instead based on direct estimation of the conditional quantile function (CQF). Since the CDF and the CQF are generalized inverses of each other, indirect estimates of the CQF and the CDF may be obtained by taking the generalized inverse of the direct estimates obtained from either approach, possibly after rearranging to guarantee monotonicity of estimated CDFs and CQFs. The equivalence between the two approaches holds for standard nonparametric estimators in the unconditional case. In the conditional case, when modeling assumptions are introduced to avoid curse-of-dimensionality problems, this equivalence is generally lost as a convenient parametric model for the CDF need not imply a convenient parametric model for the CQF, and vice versa. Despite the vast literature on the quantile regression approach, and the recent attention to the distribution regression approach, no systematic comparison of the two has been carried out yet. Our paper fills-in this gap by comparing the asymptotic properties of estimators obtained from the two approaches, both when the assumed parametric models on which they are based are correctly specified and when they are not.