Quantile Spacings: A Simple Method for the Joint Estimation of Multiple Quantiles Without Crossing

Abstract

We propose a simple but flexible parametric method for estimating multiple conditional quantiles. By construction, the estimated quantiles will satisfy the monotonicity requirement which must hold for any distribution, so, in contrast to many benchmark methods, they are not susceptible to the well-known quantile crossing problem. Rather than directly modeling the level of each individual quantile, we begin with a single quantile (usually the median), and then add or subtract sums of nonnegative functions (quantile spacings) to obtain the other quantiles. Our approach is thus a natural extension of the location-scale paradigm that also permits higher order moments (e.g., skewness and kurtosis) to vary. We propose two estimation methods and characterize the limiting behavior of each, establishing consistency, asymptotic normality, and the validity of bootstrap inference. The latter method, under an additional "linear index" assumption, respects monotonicity but preserves the computational tractability of standard linear quantile regression. We propose a simple interpolation method which generates a mapping from a finite number of quantiles to a probability density function. Simulation exercises demonstrate that the estimators perform well in finite samples. Finally, three applications demonstrate the utility of the method in time-series (forecasting), cross-sectional, and panel settings.

Keywords: Quantile Regression, Quantile Crossing Problem

JEL Classification: C14, C21, C22, C23, C51. C53