# Quantile Regression

Quantile regression allows one to estimate and conduct inference about the conditional quantile functions. It is somewhat analogous to ordinary least squares, which instead concerns the conditional mean.

## Quantiles

Let $Y$ be a random variable with cumulative distribution function $F\left(y\right)=\mathrm{Pr}\left(Y\le y\right)$. For $0\le \tau <1$, the $\tau$-th quantile of $Y$ is defined as

(1)$Q\left(\tau \right)=\mathrm{inf}\left\{y:F\left(y\right)\ge \tau \right\}.$

$Q\left(\tau \right)$ is called the quantile function. Like the distribution function, it provides a complete characterization of $Y$. The median, $Q\left(0.5\right)$ is a special case.

## Regression

The loss function in the quantile regression framework is the so called “check function”

(2)${\rho }_{\tau }\left(x\right)=\left\{\begin{array}{ll}\tau x,& \text{if}\phantom{\rule{1em}{0ex}}x\ge 0\\ \left(\tau -1\right)x,& \text{if}\phantom{\rule{1em}{0ex}}x>0\end{array}.$

For some parametric function $q\left({x}_{i},\beta \right)$, the quantile regression estimate $\stackrel{^}{\beta }$ of $\beta$ minimizes the objective function

(3)$\frac{1}{n}\sum _{i=1}^{n}{\rho }_{\tau }\left({y}_{i}-q\left({x}_{i},\beta \right)\right).$

## References

• Koenker, R., and G. W. Bassett (1978). Regression Quantiles. Econometrica 46, 33–50.