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

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 an acceptable solution but no simple and general solution has been found to date. This paper describes an elegant solution to the problem which is based on a single mathematical equation that is easy to understand and implement in R and Python, while greatly reducing the crossing problem. 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.