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.