Solution to the Non-Monotonicity and Crossing Problems in Quantile Regression

TL;DR

This paper introduces a novel, easy-to-implement method using a flexible check function to effectively eliminate quantile crossing issues in quantile regression, enhancing its reliability in econometrics and machine learning.

Contribution

It presents a unique solution to the longstanding quantile crossing problem using a flexible check function, improving estimation accuracy and interpretability.

Findings

Reduces or eliminates quantile crossing in regression models

Applicable in econometrics and machine learning contexts

Provides insights into the root causes of crossing issues

Abstract

This paper proposes a new method to address the long-standing problem of lack of monotonicity in estimation of the conditional and structural quantile function, also known as quantile crossing problem. Quantile regression is a very powerful tool in data science in general and econometrics in particular. Unfortunately, the crossing problem has been confounding researchers and practitioners alike for over 4 decades. Numerous attempts have been made to find a simple and general solution. This paper describes a unique and elegant solution to the problem based on a flexible check function that is easy to understand and implement in R and Python, while greatly reducing or even eliminating the crossing problem entirely. It will be very important in all areas where quantile regression is routinely used and may also find application in robust regression, especially in the context of machine learning. From this perspective, we also utilize the flexible check function to provide insights into the root causes of the crossing problem.