CONFIDENCE BANDS IN QUANTILE REGRESSION

Abstract

Let ( X 1 , Y 1 ), …, ( X n , Y n ) be independent and identically distributed random variables and let l ( x ) be the unknown p -quantile regression curve of Y conditional on X . A quantile smoother l n ( x ) is a localized, nonlinear estimator of l ( x ). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process { l n ( x ) – l ( x )}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup 0≤ x ≤1 | l n ( x ) − l ( x )|. The derived result helps in the construction of a uniform confidence band for the quantile curve l ( x ). This confidence band can be applied as a econometric model check. An economic application considers the relation between age and earnings in the labor market by means of parametric model specification tests, which presents a new framework to describe trends in the entire wage distribution in a parsimonious way.