Quantile regression
- Including median, minimization of sums of absolute deviations
- Koenker and Bassett or bootstrapped standard errors
Stata’s qreg command fits quantile (including median)
regression models, also known as least-absolute value (LAV) models, minimum
absolute deviation (MAD) models, and L1-norm models.
Median regression estimates the median of the dependent variable,
conditional on the values of the independent variable. This is similar to
least-squares regression, which estimates the mean of the dependent
variable. Said differently, median regression finds the regression plane
that minimizes the sum of the absolute residuals rather than the sum of the
squared residuals.
. webuse auto
(1978 Automobile Data)
. qreg price weight length foreign
Iteration 1: WLS sum of weighted deviations = 112795.66
Iteration 1: sum of abs. weighted deviations = 111901
Iteration 2: sum of abs. weighted deviations = 110529.43
Iteration 3: sum of abs. weighted deviations = 109524.57
Iteration 4: sum of abs. weighted deviations = 109468.3
Iteration 5: sum of abs. weighted deviations = 109105.27
note: alternate solutions exist
Iteration 6: sum of abs. weighted deviations = 108931.02
Iteration 7: sum of abs. weighted deviations = 108887.4
Iteration 8: sum of abs. weighted deviations = 108822.59
Median regression Number of obs = 74
Raw sum of deviations 142205 (about 4934)
Min sum of deviations 108822.6 Pseudo R2 = 0.2347
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 3.933588 .8602183 4.57 0.000 2.217937 5.649239
length | -41.25191 28.8693 -1.43 0.157 -98.82991 16.32609
foreign | 3377.771 577.3391 5.85 0.000 2226.305 4529.237
_cons | 344.6494 3260.244 0.11 0.916 -6157.702 6847.001
------------------------------------------------------------------------------
By default, qreg performs median regression—the estimates above
were obtained by minimizing the sums of the absolute residuals.
By comparison, the results from least-squares regression are
. regress price weight length foreign
Source | SS df MS Number of obs = 74
-------------+------------------------------ F( 3, 70) = 28.39
Model | 348565467 3 116188489 Prob > F = 0.0000
Residual | 286499930 70 4092856.14 R-squared = 0.5489
-------------+------------------------------ Adj R-squared = 0.5295
Total | 635065396 73 8699525.97 Root MSE = 2023.1
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 5.774712 .9594168 6.02 0.000 3.861215 7.688208
length | -91.37083 32.82833 -2.78 0.007 -156.8449 -25.89679
foreign | 3573.092 639.328 5.59 0.000 2297.992 4848.191
_cons | 4838.021 3742.01 1.29 0.200 -2625.183 12301.22
------------------------------------------------------------------------------
qreg can also estimate the regression plane for quantiles other than
the 0.5 (median). For instance, the following model describes the 25th
percentile (.25 quantile) of price:
. qreg price weight length foreign, quantile(.25)
Iteration 1: WLS sum of weighted deviations = 98938.466
Iteration 1: sum of abs. weighted deviations = 99457.766
Iteration 2: sum of abs. weighted deviations = 91339.779
Iteration 3: sum of abs. weighted deviations = 86833.291
Iteration 4: sum of abs. weighted deviations = 83894.441
Iteration 5: sum of abs. weighted deviations = 82186.049
Iteration 6: sum of abs. weighted deviations = 75246.848
Iteration 7: sum of abs. weighted deviations = 71442.907
Iteration 8: sum of abs. weighted deviations = 70452.616
Iteration 9: sum of abs. weighted deviations = 69646.639
Iteration 10: sum of abs. weighted deviations = 69603.554
.25 Quantile regression Number of obs = 74
Raw sum of deviations 83825.5 (about 4187)
Min sum of deviations 69603.55 Pseudo R2 = 0.1697
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 1.831789 .668093 2.74 0.008 .4993194 3.164258
length | 2.845558 24.78057 0.11 0.909 -46.57773 52.26885
foreign | 2209.925 434.631 5.08 0.000 1343.081 3076.769
_cons | -1879.775 2808.067 -0.67 0.505 -7480.287 3720.737
------------------------------------------------------------------------------
The following describes the 75th percentile (.75 quantile):
. qreg price weight length foreign, quantile(.75)
Iteration 1: WLS sum of weighted deviations = 110931.48
Iteration 1: sum of abs. weighted deviations = 111305.91
Iteration 2: sum of abs. weighted deviations = 105989.57
Iteration 3: sum of abs. weighted deviations = 100378.89
Iteration 4: sum of abs. weighted deviations = 99796.49
Iteration 5: sum of abs. weighted deviations = 98796.212
Iteration 6: sum of abs. weighted deviations = 98483.669
Iteration 7: sum of abs. weighted deviations = 98395.935
.75 Quantile regression Number of obs = 74
Raw sum of deviations 159721.5 (about 6342)
Min sum of deviations 98395.94 Pseudo R2 = 0.3840
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 9.22291 2.653579 3.48 0.001 3.930515 14.51531
length | -220.7833 80.13907 -2.76 0.007 -380.6156 -60.95096
foreign | 3595.133 1727.704 2.08 0.041 149.3355 7040.931
_cons | 20242.9 8534.529 2.37 0.020 3221.323 37264.49
------------------------------------------------------------------------------
In addition to Koenker and Basset standard errors, Stata can provide
bootstrapped standard errors, using the
bsqreg command:
. set seed 1001
. bsqreg price weight length foreign
(fitting base model)
(bootstrapping ....................)
Median regression, bootstrap(20) SEs Number of obs = 74
Raw sum of deviations 142205 (about 4934)
Min sum of deviations 108822.6 Pseudo R2 = 0.2347
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 3.933588 3.12446 1.26 0.212 -2.297951 10.16513
length | -41.25191 83.71266 -0.49 0.624 -208.2116 125.7077
foreign | 3377.771 1040.209 3.25 0.002 1303.14 5452.402
_cons | 344.6494 7053.301 0.05 0.961 -13722.72 14412.01
------------------------------------------------------------------------------
The coefficient estimates are the same as those in the first example. The
standard errors, and, therefore, the t statistics, significance levels, and
confidence intervals differ.
Stata can also perform simultaneous-quantile regression. Previously, we ran
separate regressions for the .25, .5, and .75 quantiles. With
simultaneous-quantile regression, we can estimate all the effects
simultaneously:
. set seed 1001
. sqreg price weight length foreign, q(.25 .5 .75)
(fitting base model)
(bootstrapping ....................)
Simultaneous quantile regression Number of obs = 74
bootstrap(20) SEs .25 Pseudo R2 = 0.1697
.50 Pseudo R2 = 0.2347
.75 Pseudo R2 = 0.3840
------------------------------------------------------------------------------
| Bootstrap
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
q25 |
weight | 1.831789 .9837745 1.86 0.067 -.1302874 3.793865
length | 2.845558 23.78166 0.12 0.905 -44.58547 50.27658
foreign | 2209.925 791.2162 2.79 0.007 631.8944 3787.956
_cons | -1879.775 2862.855 -0.66 0.514 -7589.559 3830.01
-------------+----------------------------------------------------------------
q50 |
weight | 3.933588 2.800286 1.40 0.165 -1.651407 9.518583
length | -41.25191 76.813 -0.54 0.593 -194.4506 111.9468
foreign | 3377.771 1099.259 3.07 0.003 1185.369 5570.173
_cons | 344.6494 6764.42 0.05 0.960 -13146.56 13835.86
-------------+----------------------------------------------------------------
q75 |
weight | 9.22291 2.903055 3.18 0.002 3.43295 15.01287
length | -220.7833 104.5321 -2.11 0.038 -429.2661 -12.30049
foreign | 3595.133 1051.723 3.42 0.001 1497.539 5692.728
_cons | 20242.9 11054.76 1.83 0.071 -1805.125 42290.93
------------------------------------------------------------------------------
We can test whether the effect of weight is the same at the 25th and 75th
percentiles:
. test[q25]weight = [q75]weight
( 1) [q25]weight - [q75]weight = 0
F( 1, 70) = 7.87
Prob > F = 0.0065
We can obtain a confidence interval for the difference in the effect of
weight at the 25th and 75th percentiles:
. lincom [q75]weight-[q25]weight
( 1) - [q25]weight + [q75]weight = 0
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | 7.391121 2.634483 2.81 0.006 2.13681 12.64543
------------------------------------------------------------------------------
Stata also performs interquantile regression, which focuses on one quantile
comparison:
. set seed 1001
. iqreg price weight length foreign, q(.25 .75)
(fitting base model)
(bootstrapping ....................)
.75-.25 Interquantile regression Number of obs = 74
bootstrap(20) SEs .75 Pseudo R2 = 0.3840
.25 Pseudo R2 = 0.1697
------------------------------------------------------------------------------
| Bootstrap
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | 7.391121 2.634483 2.81 0.006 2.13681 12.64543
length | -223.6288 98.00503 -2.28 0.026 -419.0937 -28.16397
foreign | 1385.208 1080.812 1.28 0.204 -770.4034 3540.819
_cons | 22122.68 10940.56 2.02 0.047 302.4109 43942.95
------------------------------------------------------------------------------
References
- Gould, W. 1992.
-
sg11.1: Quantile regression with bootstrapped standard errors.
Stata Technical Bulletin
9: 19–21. Reprinted in Stata Technical Bulletin Reprints, vol. 2,
pp. 137–150.
- Gould, W., and W. H. Rogers. 1994.
-
Quantile regression as an alternative to robust regression.
Proceedings of the Statistical Computing Section.
Alexandria, VA: American Statistical Association.
- Rogers, W. H. 1992.
-
sg11: Quantile regression standard errors.
Stata Technical Bulletin 9: 16–19.
Reprinted in Stata Technical Bulletin Reprints, vol. 2, pp. 133–137.
- ------. 1993.
-
sg11.2: Calculation of quantile regression standard errors.
Stata Technical Bulletin 13: 18–19.
Reprinted in Stata Technical Bulletin,
vol. 3, pp. 77–78.
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