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, Econometrica 57, 1403–1430). The AQD summarizes the marginal response of the covariates on the CQF and defines a nonparametric quantile regression coefficient. In semiparametric single-index and partially 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 by the entire support of the covariates. To facilitate 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 nonlinear functionals of the quantile processes.
Highlights
The weighted average quantile derivative (AQD) is the weighted expected value of the partial derivatives of the conditional quantile function (CQF), defined as βW (τ ) ≡ E [∇Q(τ |X)W(X)], (1)where Q(τ |X) is the τ th CQF of the dependent variable Y given the continuous covariates X
The AQD defines a nonparametric quantile regression (QR) parameter that summarizes the marginal effect of X on the τ th CQF
We propose two-step estimators for three estimands: the weighted average quantile response (AQR) βφ(τ ) in (3), the weighted AQD βW (τ ) in (2), and the density-weighted AQD
Summary
The weighted average quantile derivative (AQD) is the weighted expected value of the partial derivatives of the conditional quantile function (CQF), defined as βW (τ ) ≡ E [∇Q(τ |X)W(X)],. The third result is a Bahadur-type linear representation of the CQF estimator that is uniform over values of the covariates in a sequence of expanding compact interior support and over quantile levels in a compact subset of (0,1). For the local polynomial estimator of the CQF, Chaudhuri et al (1997) and Kong, Linton, and Xia (2010) derive Bahadur representations for uniformity in the covariates X, Qu and Yoon’s (2015) result is uniform over the quantile τ , and Guerre and Sabbah (2012) and Fan and Guerre (2016) provide the uniformity in X and τ To extend their Bahadur representations to uniformity on expanding interior supports, the uniform convergence rate is penalized by the lower bound of the density at a slower rate, as noted in Hansen (2008).
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