# Quantile regression estimator formula

I have seen two different representations of the quantile regression estimator which are

$$Q(\beta_{q}) = \sum^{n}_{i:y_{i}\geq x'_{i}\beta} q\mid y_i - x'_i \beta_q \mid + \sum^{n}_{i:y_{i}< x'_{i}\beta} (1-q)\mid y_i - x'_i \beta_q \mid$$

and $$Q(\beta_q) = \sum^{n}_{i=1} \rho_q (y_i - x'_i \beta_q), \hspace{1cm} \rho_q(u) = u_i(q - 1(u_i < 0 ))$$

where $u_i = y_i - x'_i \beta_q$. Can somebody tell me how to show equivalence of these two expressions? Here is what I tried so far, starting from the second expression.

\begin{align} Q(\beta_q) &= \sum^{n}_{i=1} u_i(q - 1(u_i < 0 )) (y_i - x'_i \beta_q) \newline &= \sum^{n}_{i=1}(y_i - x'_i \beta_q)(q - 1(y_i - x'_i \beta_q < 0 )) (y_i - x'_i \beta_q) \newline &=\left[ \sum^{n}_{i:y_{i}\geq x'_{i}\beta}(q(y_i - x'_i\beta_q)) + \sum^{n}_{i:y_{i}< x'_{i}\beta}(q(y_i - x'_i\beta_q)-(y_i - x'_i\beta_q)) \right](y_i - x'_i\beta_q) \end{align} But from this point I got stuck on how to proceed. Please not that this is not a homework or assignment question. Many thanks.

If you remember, OLS minimizes the sum of the squared residuals $\sum_i u_{i}^{2}$ whereas median regression minimizes the sum of absolute residuals $\sum_i \mid u_i \mid$. The median or least absolute deviations (LAD) estimator is a special case of quantile regression in which you have $q = .5$. In quantile regression we minimize a sum of absolute errors that receives asymmetric weights for overprediction $(1-q)$ and $q$ for underprediction. You can start from the LAD representation and extend this as the sum of the fraction of the data which are weighted by $q$ and $(1-q)$ given their value of $u_i$, and work on it as follows:
\begin{align} \rho_q(u) &= 1(u_i>0) \, q\mid u_i\mid + 1(u_i\leq 0) \, (1-q)\mid u_i \mid \newline &= 1(y_i - x'_i \beta_q > 0) \, q\mid y_i - x'_i \beta_q \mid + 1(y_i - x'_i\beta_q \leq 0) \, (1-q)\mid y_i - x'_i \beta_q \mid \end{align} This just uses the fact that $u_i = y_i - x'_i \beta_q$ and then you can re-write the indicator function as sums of the observations that satisfy the conditions of the indicators. This will give the first expression you wrote down for the quantile regression estimator.
\begin{align} &= \sum^{n}_{i:y_i>x'_i\beta_q}q\mid y_i - x'_i\beta_q \mid + \sum^{n}_{i:y_i\leq x'_i\beta_q} (1-q) \mid y_i - x'_i\beta_q \mid \newline &= q \sum^{n}_{i:y_i>x'_i\beta_q} \mid y_i - x'_i\beta_q \mid + (1-q)\sum^{n}_{i:y_i\leq x'_i\beta_q} \mid y_i - x'_i\beta_q \mid \newline &= q \sum^{n}_{i:y_i>x'_i\beta_q} (y_i - x'_i\beta_q) - (1-q)\sum^{n}_{i:y_i\leq x'_i\beta_q} ( y_i - x'_i\beta_q ) \newline &= q \sum^{n}_{i:y_i>x'_i\beta_q} (y_i - x'_i\beta_q) - \sum^{n}_{i:y_i\leq x'_i\beta_q} (y_i - x'_i\beta_q) + q \sum^{n}_{i:y_i\leq x'_i\beta_q} (y_i - x'_i\beta_q) \newline &= q \sum^{n}_{i=1} (y_i - x'_i \beta_q) - \sum^{n}_{i=1}1(y_i - x'_i\beta_q\leq 0)(y_i - x'_i\beta_q) \newline &= \sum^{n}_{i=1}(q - 1(u_i \leq 0))u_i \end{align}
The second line takes out the weights from the summations. The third line gets rid of the absolute values and replaces them by the actual values. By definition $y_i - x'_i\beta_q$ is negative whenever $y_i < x'_i\beta_q$, hence the sign change in this line. The fourth line multiplies out $(1-q)$. You then realize that $$q\sum^{n}_{i:y_i>x'_i\beta_q}(y_i - x'_i\beta_q) + q\sum^{n}_{i:y_i \leq x'_i\beta_q}(y_i - x'_i\beta_q) = \sum^{n}_{i=1}(y_i - x'_i\beta_q)$$ and replacing the summation of the middle term in the fourth line by the corresponding indicator you arrive at the fifth line. Factorizing and then replacing $y_i - x'_i\beta_q$ with $u_i$ yields the second expression of your estimator.