qreg - Stata

qreg — Quantile regression 7
Example 1: Estimating the conditional median
Consider a two-group experimental design with 5 observations per group:
. use http://www.stata-press.com/data/r13/twogrp
. list
x
y
1. 0 0
2. 0 1
3. 0 3
4. 0 4
5. 0 95
6. 1 14
7. 1 19
8. 1 20
9. 1 22
10. 1 23
. qreg y x
Iteration 1: WLS sum of weighted deviations = 121.88268
Iteration 1: sum of abs. weighted deviations = 111
Iteration 2: sum of abs. weighted deviations = 110
Median regression Number of obs = 10
Raw sum of deviations 157 (about 14)
Min sum of deviations 110 Pseudo R2 = 0.2994
y Coef. Std. Err. t P>|t| [95% Conf. Interval]
x 17 18.23213 0.93 0.378 -25.04338 59.04338
_cons 3 12.89207 0.23 0.822 -26.72916 32.72916
We have estimated the equation
y median = 3 + 17 x
We look back at our data. x takes on the values 0 and 1, so the median for the x = 0 group is 3,
whereas for x = 1 it is 3 + 17 = 20. The output reports that the raw sum of absolute deviations about
14 is 157; that is, the sum of |y − 14| is 157. Fourteen is the unconditional median of y, although
in these data, any value between 14 and 19 could also be considered an unconditional median (we
have an even number of observations, so the median is bracketed by those two values). In any case,
the raw sum of deviations of y about the median would be the same no matter what number we
choose between 14 and 19. (With a “median” of 14, the raw sum of deviations is 157. Now think
of choosing a slightly larger number for the median and recalculating the sum. Half the observations
will have larger negative residuals, but the other half will have smaller positive residuals, resulting in
no net change.)
We turn now to the actual estimated equation. The sum of the absolute deviations about the
solution y median = 3 + 17x is 110. The pseudo-R 2 is calculated as 1 − 110/157 ≈ 0.2994. This
result is based on the idea that the median regression is the maximum likelihood estimate for the
double-exponential distribution.