Quantile and Probability Curves Without Crossing

Abstract

This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem (Bassett and Koenker (1982)). The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural distribution and quantile functions using data on Vietnam veteran status and earnings.