Fundamentals of Quantile Regression

Abstract

In this chapter, we seek to provide a basic conceptual guide to quantile regression, illustrating the ideas with a number of examples and stressing various aspects of the interpretation of quantile regression. We begin with a discussion of quantile treatment effects in the two-sample treatment-control model. In this context, the difference between the empirical quantile functions of the treatment and control observations provides a natural generalization of conventional mean measures of the treatment effect. This “quantile treatment effect” is precisely what is delivered by the the quantile regression estimator of a model with a single binary indicator variable. We describe some basic characteristics of the quantile regression estimator, its equivariance properties, and robustness. The interpretation of estimated quantile regression parameters is described in the context of several applications. Some issues of misspecification are raised, and the chapter concludes with an interpretation of the quantile regression model as a random coefficient model. QUANTILE TREATMENT EFFECTS The simplest formulation of regression is the classical two-sample treatment-control model. We begin by reconsidering a general model of two-sample treatment response introduced by Lehmann and Doksum in the 1970s. This model provides a natural introduction to the interpretation of quantile regression models in more general settings. Lehmann (1974) proposed the following model of treatment response: Suppose the treatment adds the amount Δ( x ) when the response of the untreated subject would be x . Then the distribution G of the treatment responses is that of the random variable X + Δ( X ) where X is distributed according to F .