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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 $-14.96$

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  • $\begingroup$ Use a -1 in the formula to remove the intercept. E.g. income ~ foodexp-1 This is the same for lm and rq. $\endgroup$ Jul 6 at 14:40
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    $\begingroup$ 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 $y$. $\endgroup$ Jul 6 at 15:02
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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.

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  • $\begingroup$ 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? $\endgroup$ Jul 6 at 15:06
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    $\begingroup$ Not that I know of, which is not to say it is impossible. $\endgroup$
    – Peter Flom
    Jul 6 at 16:00
  • $\begingroup$ @PeterFlom: Please see request for information about CV.SE work here. $\endgroup$
    – Ben
    Aug 27 at 6:37

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