Nonparametric Weighted Average Quantile Derivative

Abstract

The weighted Average Quantile Derivative (AQD) is the expected value of the partial derivative of the conditional quantile function (CQF) weighted by a function of the covariates. We consider two weighting functions: a known function chosen by researchers and the density function of the covariates that is parallel to the average mean derivative in Powell, Stock, and Stoker (1989). The AQD summarizes the marginal response of the covariates on the CQF and defines a nonparametric quantile regression coefficient. In semiparametric single-index and partial linear models, the AQD identifies the coefficients up to scale. In nonparametric nonseparable structural models, the AQD conveys an average structural effect under certain independence assumptions. Including a stochastic trimming function, the proposed two-step estimator is root-n-consistent for the AQD defined on the entire support of the covariates. To facilitate our tractable asymptotic analysis, a key preliminary result is a new Bahadur-type linear representation of the generalized inverse kernel-based CQF estimator uniformly over the covariates in an expanding compact set and over the quantile levels. The weak convergence to Gaussian processes applies to the differentiable non-linear functionals of the quantile processes.

Keywords: average derivative, conditional quantile function, nonparametric kernel estimation, stochastic trimming, Bahadur representation

JEL Classification: C13, C14, C21