Which Quantile is the Most Informative ? Maximum Entropy Quantile Regression ∗

This paper studies the connections among quantile regression, the asymmetric Laplace distribution, and the maximum entropy. We show that the maximum likelihood problem is equivalent to the solution of a maximum entropy problem where we impose moment constraints given by the joint consideration of the mean and median. Using the resulting score functions we develop a maximum entropy quantile regression estimator. This approach delivers estimates for the slope parameters together with the associated “most informative” quantile. Similarly, this method can be seen as a penalized quantile regression estimator, where the penalty is given by deviations from the median regression. We derive the asymptotic properties of this estimator by showing consistency and asymptotic normality under certain regularity conditions. Finally, an application to the U.S. wage data to evaluate the effect of training on wages illustrates the usefulness and implementation of our methodology.