  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.
