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-t employing the theory of Bessel processes as in Kiefer (1959). However, it is fre-quently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as \ 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 mar-tingale 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 ex-periments, conducted in 1988-89, were designed to test the ecacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells. 1.