Conditional Quantile Processes Based on Series or Many Regressors

Abstract

Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR-series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from the data.We develop large sample theory for the QR-series coefficient process, namely we obtain uniform strong approximations to the QR-series coefficient process by conditionally pivotal and Gaussian processes. Based on these two strong approximations, or couplings, we develop four resampling methods (pivotal, gradient bootstrap, Gaussian, and weighted bootstrap) that can be used for inference on the entire QR-series coefficient function.We apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function, such as the conditional quantile function itself, its partial derivatives, average partial derivatives, and conditional average partial derivatives.Specifically, we obtain uniform rates of convergence and show how to use the four resampling methods mentioned above for inference on the functionals. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and the covariate value, and covering the pointwise case as a by-product. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function and test the Slutsky condition of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption. and Yale for many useful suggestions. We are grateful to Adonis Yatchew for giving us permission to use the data set in the empirical application. We gratefully acknowledge research support from the NSF. The R package quantreg.nonpar implements some of the methods of this paper [66]. 1 arXiv:1105.6154v4 [stat.ME] 9 Aug 2018 for a careful and detailed description of these series functions; see also Belloni et al [10] for an overview of recent advances on series approximating functions.We define the QR-series approximating function x → Z(x) β(u) mapping X into R, and, for all x ∈ X , the QR-series approximation error2 The optimization problem (2.2) has a finite solution if E[|Q(u, X)|] is finite. In addition, since the function z → ρu(z) is strictly convex, the solution is unique if the matrix E[Z(X)Z(X) ] is non-singular, which is assumed in Condition S below. The term ρu(Y − Q(u, X)) does not affect the optimization problem but guarantees the existence of the solution when E[|Y |] is not finite.3 Interestingly, in the case of B-splines and compactly supported wavelets, the entire collection of series terms is dependent upon th...

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