The QUANTREG Procedure

Quantile Regression as an Optimization Problem

The model for linear quantile regression is

$\text{[math]}$

where $\text{[math]}$ is the $\text{[math]}$ vector of responses, $\text{[math]}$ is the $\text{[math]}$ regressor matrix, $\text{[math]}$ is the $\text{[math]}$ vector of unknown parameters, and $\text{[math]}$ is the $\text{[math]}$ vector of unknown errors.

$\text{[math]}$ regression, also known as median regression, is a natural extension of the sample median when the response is conditioned on the covariates. In $\text{[math]}$ regression, the least absolute residuals estimate $\text{[math]}$ , referred to as the $\text{[math]}$ -norm estimate, is obtained as the solution of the minimization problem

$\text{[math]}$

More generally, for quantile regression Koenker and Bassett (1978) defined the $\text{[math]}$ th regression quantile, $\text{[math]}$ , as any solution to the minimization problem

$\text{[math]}$

The solution is denoted as $\text{[math]}$ , and the $\text{[math]}$ -norm estimate corresponds to $\text{[math]}$ . The $\text{[math]}$ th regression quantile is an extension of the $\text{[math]}$ th sample quantile $\text{[math]}$ , which can be formulated as the solution of

$\text{[math]}$

If you specify weights $\text{[math]}$ , with the WEIGHT statement, weighted quantile regression is carried out by solving

$\text{[math]}$

Weighted regression quantiles $\text{[math]}$ can be used for L-estimation; refer to Koenker and Zhao (1994).