Goodness of fit and related inference processes for quantile regression

We introduce a goodness-of-fit process for quantile regression analogous to the conventional R² statistic of least squares regression. Several related inference processes designed to test composite hypotheses about the combined effect of several covariates over an entire range of conditional quantile functions are also formulated. The asymptotic behavior of the inference processes is shown to be closely related to earlier p-sample goodness-of-fit theory involving Bessel processes. The approach is illustrated with some hypothetical examples, an application to recent empirical models of international economic growth, and some Monte Carlo evidence.