Likelihood Ratio and Goodness of Fit Processes for Quantile Regression

We consider several variants of a likelihood-ratio process for quantile regression designed to test composite hypotheses about the combined influence of several covariates over an entire range of conditional quantile functions. A closely related process is proposed as a goodness-of-fit criterion for quantile regression and some analogies to rank tests, conventional R-squared, and Gaussian likelihood-ratios are developed. The approach is illustrated with several hypothetical examples, an application to recent empirical models of international growth, and some Monte-Carlo evidence