Inference on the quantile regression process

Tests based on the quantile regression process can be formulated like the classical Kolmogorov-Smirnov and Cramer-von-Mises tests of goodness-of-fit employing the theory of Bessel processes as in Kiefer (1959). However, it is frequently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as "the Durbin problem" since it was posed in Durbin (1973), for parametric empirical processes. In this paper we consider an approach to the Durbin problem involving a martingale transformation of the parametric empirical process suggested by Khmaladze (1981) and show that it can be adapted to a wide variety of inference problems involving the quantile regression process. In particular, we suggest new tests of the location shift and location-scale shift models that underlie much of classical econometric inference. The methods are illustrated with a reanalysis of data on unemployment durations from the Pennsylvania Reemployment Bonus Experiments. The Pennsylvania experiments, conducted in 1988-89, were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells.