I would like to please ask for your help concerning the following issue.
After consecutively running two separate quantile regressions for percentiles $p_i$ and $p_j$ with $j>i$ [e.g., for the first percentile, $p_1$, and the second percentile, $p_2$ ($i=1,~j=2$)], are we "guaranteed" to obtain $\hat{p}_j \ge \hat{p}_i$, i.e., will the predicted value for the higher percentile always be larger or equal to the predicted value for the lower percentile? For example, will $\hat{p}_2 \ge \hat{p}_1$ "always" be true?
As a quick-and-dirty exploration anyone as dull and clueless as me is restrained to, I simulated some data in Stata and used the qreg and predict commands, whose output suggest that this is not the case, which, however, for obvious reasons, does not constitute a disprove.
I am using quotation marks to indicate my handwavyness -- I am mainly dealing with linear combinations of real random variables.
