Take a dataset and suppose we fit two quantile regression models to it, one with the raw dependent variable (DV) and one with the logged DV. Then look at each model's predictions for the training data, reversing the transformation in the case of the logged model. Here's an example in R:
library(quantreg) set.seed(1) x1 = rnorm(100) y = exp(x1 + 3*rnorm(100)) m = rq(y ~ x1, tau = .5) p1 = predict(m) m = rq(log(y) ~ x1, tau = .5) p2 = exp(predict(m)) print(head(p1)) print(head(p2))
The two models give different predictions:
1 2 3 4 5 6 0.4631787 0.8619277 0.3602179 1.5567692 0.9337257 0.3676802 1 2 3 4 5 6 0.2776789 0.7859069 0.2122640 4.8162168 0.9478212 0.2164372
But how can this be? The models are fit in terms of the quantiles of the DV, and the natural logarithm is an increasing function, so it preserves quantile ranks.