# How to obtain a 0 intercept in quantile regression

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 $$-14.96$$ • Use a -1 in the formula to remove the intercept. E.g. income ~ foodexp-1 This is the same for lm and rq. Jul 6, 2021 at 14:40 • Thank you for your answer. What you propose is to fit a model without the intercept, but that is not what I am asking. If I fit a model without intercept, the slope will be different. What I want is to understand how the intercept in quantile regression is related to the data. Eg, in least squares regression the intercept is related to the mean value of$X$and$y$. Jul 6, 2021 at 15:02 ## 1 Answer You can simply add 14.96 to income: income_med_adj <- engel$$income + 14.96 rq(income_med_adj ~ foodexp, data=engel, tau=0.5)$$coefficients %>% round(3)  Of course, this would be different for different quantiles. • Thank you for your answer. That would solve the specific example I posed, but I would be interested in a more general explanation. In the lm model I can have a model that has 0 intercept without solving first the model with the intercept and then substracting, because I know that the intercept is related to the mean values of$X$and$y\$. Is there something similar in quantile regression? Jul 6, 2021 at 15:06
• Not that I know of, which is not to say it is impossible. Jul 6, 2021 at 16:00