Quantile regression models are a type of models that provide estimates of the quantiles of a response variable $y$ given a set of covariates $X$ in the form of a linear equation such as
$$ y = \beta_0 + \beta_1 x_1 + \ldots \beta_p x_p$$
where the $\beta$ coefficients vary depending on the quantile for which the model is solved. My question is: what are the requirements under which $\beta_0=0$?
For example, if instead of a quantile regression model we were solving a ordinary least squares linear regression model, I would know that, if both $y$ and $X$ are centered, then $\beta_0$ would be 0. Is there something equivalent for quantile regression?
See the following example:
library(quantreg) library(tidyverse) # Load the data and solve a linear regression model data(engel) lm(income ~ foodexp, data=engel)$coefficients %>% round(3) (Intercept) foodexp -85.736323 1.711462 # Center the data and solve the model again # Here the intercept takes value 0 and the slope is the same as before engel_scaled = scale(engel, scale=F) %>% data.frame() lm(income ~ foodexp, data=engel_scaled)$coefficients %>% round(3) (Intercept) foodexp 0.000 1.711 # Now using a quantile regression model. rq(income ~ foodexp, data=engel, tau=0.5)$coefficients %>% round(3) (Intercept) foodexp -14.961 1.548
What transformation would we need to do on $y$ and $X$ so that the intercept of the quantile regression model is $0$ instead of $-14.96$