On assessing model adequacy in linear quantile regression

Noh, Hohsuk [UCL] El Ghouch, Anouar [UCL] Van Keilegom, Ingrid [UCL]

In this paper we consider (possibly misspecified) linear quantile regression models, and study a measure for the quality-of-fit of these models, (a version of which has been) previously proposed by Koenker and Machado (1999). The measure is based on an adaptation to quantile regression of the famous coefficient of determination originally proposed for mean regression, and compares a ‘reduced’ model to a ‘full’ model, both of which can be misspecified. We propose an estimator of this measure, and prove its asymptotic distribution both in the non-degenerate and the degenerate case. The finite sample performance of the estimator is studied through a number of simulation experiments. The proposed measure is also applied to a data set on body fat measures.