Quantile Regression: A Simplified Approach to a Goodness-of-fit Test

Recently, He and Zhu (2003) derived an omnibus goodness-of-fit test for linear or nonlinear quantile regression models based on a CUSUM process of the gradient vector, and they suggested using a particular simulation method for determining critical values for their test statistic. But despite the speed of modern computers, execution time can be high. One goal in this note is to suggest a slight modification of their method that eliminates the need for simulations among a collection of important and commonly occurring situations. For a broader range of situations, the modification can be used to determine a critical value as a function of the sample size (n), the number of predictors (q), and the quantile of interest (γ). This is in contrast to the He and Zhu approach where the critical value is also a function of the observed values of the q predictors. As a partial check on the suggested modification in terms of controlling the Type I error probability, simulations were performed for the same situations considered by He and Zhu, and some additional simulations are reported for a much wider range of situations.