Quality of Fit Measures in the Framework of Quantile Regression

In regression experiments, to learn about the strength of the relationship between a covariate vector and a dependent variable, we propose a 'coefficient of determination' based on the quantiles. Such a coefficient is a 'local' measure in the sense that the strength is measured at a prespecified quantile level. Once estimated, it can be used, for example, to measure the relative importance of a subset of covariates in the quantile regression context. Related to this coefficient, we also propose a new 'local' lack-of-fit measure of a given parametric model. We provide some asymptotic results of the proposed measures and carry out a Monte Carlo simulation study to illustrate their use and performance in practice.