Including median, minimization of sums of absolute deviations
There are now three ways to obtain the VCE:
the standard Koenker and Bassett method appropriate for i.i.d. errors;
a Huber sandwich estimator that can be used even if the errors are not i.i.d.;
the bootstrap.
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Stata 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 = 56397.829 Iteration 1: Sum of abs. weighted deviations = 55950.5 Iteration 2: Sum of abs. weighted deviations = 55264.718 Iteration 3: Sum of abs. weighted deviations = 54762.283 Iteration 4: Sum of abs. weighted deviations = 54734.152 Iteration 5: Sum of abs. weighted deviations = 54552.638 note: alternate solutions exist. Iteration 6: Sum of abs. weighted deviations = 54465.511 Iteration 7: Sum of abs. weighted deviations = 54443.699 Iteration 8: Sum of abs. weighted deviations = 54411.294 Median regression Number of obs = 74 Raw sum of deviations 71102.5 (about 4934) Min sum of deviations 54411.29 Pseudo R2 = 0.2347
| price | Coefficient Std. err. t P>|t| [95% conf. interval] | |
| weight | 3.933588 1.328718 2.96 0.004 1.283543 6.583632 | |
| length | -41.25191 45.46469 -0.91 0.367 -131.9284 49.42456 | |
| foreign | 3377.771 885.4198 3.81 0.000 1611.857 5143.685 | |
| _cons | 344.6489 5182.394 0.07 0.947 -9991.31 10680.61 | |
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 | Coefficient 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 = 49469.235 Iteration 1: Sum of abs. weighted deviations = 49728.883 Iteration 2: Sum of abs. weighted deviations = 45669.89 Iteration 3: Sum of abs. weighted deviations = 43416.646 Iteration 4: Sum of abs. weighted deviations = 41947.221 Iteration 5: Sum of abs. weighted deviations = 41093.025 Iteration 6: Sum of abs. weighted deviations = 37623.424 Iteration 7: Sum of abs. weighted deviations = 35721.453 Iteration 8: Sum of abs. weighted deviations = 35226.308 Iteration 9: Sum of abs. weighted deviations = 34823.319 Iteration 10: Sum of abs. weighted deviations = 34801.777 .25 Quantile regression Number of obs = 74 Raw sum of deviations 41912.75 (about 4187) Min sum of deviations 34801.78 Pseudo R2 = 0.1697
| price | Coefficient Std. err. t P>|t| [95% conf. interval] | |||||
| weight | 1.831789 .6328903 2.89 0.005 .5695289 3.094049 | |||||
| length | 2.84556 21.65558 0.13 0.896 -40.34514 46.03626 | |||||
| foreign | 2209.925 421.7401 5.24 0.000 1368.791 3051.059 | |||||
| _cons | -1879.775 2468.46 -0.76 0.449 -6802.963 3043.413 | |||||
Here, we perform median regression but request robust standard errors.
. qreg price weight length foreign, vce(robust) Iteration 1: WLS sum of weighted deviations = 56397.829 Iteration 1: Sum of abs. weighted deviations = 55950.5 Iteration 2: Sum of abs. weighted deviations = 55264.718 Iteration 3: Sum of abs. weighted deviations = 54762.283 Iteration 4: Sum of abs. weighted deviations = 54734.152 Iteration 5: Sum of abs. weighted deviations = 54552.638 note: alternate solutions exist. Iteration 6: Sum of abs. weighted deviations = 54465.511 Iteration 7: Sum of abs. weighted deviations = 54443.699 Iteration 8: Sum of abs. weighted deviations = 54411.294 Median regression Number of obs = 74 Raw sum of deviations 71102.5 (about 4934) Min sum of deviations 54411.29 Pseudo R2 = 0.2347
| Robust | ||||||||
| price | Coefficient std. err. t P>|t| [95% conf. interval] | |||||||
| weight | 3.933588 1.694477 2.32 0.023 .55406 7.313116 | |||||||
| length | -41.25191 51.73571 -0.80 0.428 -144.4355 61.93171 | |||||||
| foreign | 3377.771 728.5115 4.64 0.000 1924.801 4830.741 | |||||||
| _cons | 344.6489 5096.528 0.07 0.946 -9820.055 10509.35 | |||||||
Stata can provide bootstrapped standard errors, using the bsqreg command
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. set seed 1001 . bsqreg price weight length foreign (fitting base model) Bootstrap replications (20).................... Median regression, bootstrap(20) SEs Number of obs = 74 Raw sum of deviations 71102.5 (about 4934) Min sum of deviations 54411.29 Pseudo R2 = 0.2347
1 2 3 4 5
| price | Coefficient Std. err. t P>|t| [95% conf. interval] | |||||||
| weight | 3.933588 2.941839 1.34 0.186 -1.933726 9.800901 | |||||||
| length | -41.25191 73.47105 -0.56 0.576 -187.7853 105.2815 | |||||||
| foreign | 3377.771 1352.518 2.50 0.015 680.2582 6075.284 | |||||||
| _cons | 344.6489 5927.045 0.06 0.954 -11476.47 12165.77 | |||||||
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. With simultaneous-quantile regression, we can estimate multiple quantile regressions simultaneously:
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. set seed 1001 . sqreg price weight length foreign, q(.25 .5 .75) (fitting base model) Bootstrap replications (20).................... 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
1 2 3 4 5
| Bootstrap | |||||||
| price | Coefficient std. err. t P>|t| [95% conf. interval] | ||||||
| q25 | |||||||
| weight | 1.831789 1.250388 1.46 0.147 -.6620304 4.325608 | ||||||
| length | 2.84556 24.53036 0.12 0.908 -46.0787 51.76982 | ||||||
| foreign | 2209.925 1099.174 2.01 0.048 17.6916 4402.159 | ||||||
| _cons | -1879.775 3087.115 -0.61 0.545 -8036.831 4277.282 | ||||||
| q50 | |||||||
| weight | 3.933588 2.153228 1.83 0.072 -.3608896 8.228065 | ||||||
| length | -41.25191 55.61779 -0.74 0.461 -152.1781 69.67427 | ||||||
| foreign | 3377.771 1151.72 2.93 0.005 1080.738 5674.804 | ||||||
| _cons | 344.6489 5152.738 0.07 0.947 -9932.164 10621.46 | ||||||
| q75 | |||||||
| weight | 9.22291 2.315138 3.98 0.000 4.605513 13.84031 | ||||||
| length | -220.7833 83.26476 -2.65 0.010 -386.8496 -54.71695 | ||||||
| foreign | 3595.133 1072.378 3.35 0.001 1456.342 5733.924 | ||||||
| _cons | 20242.9 9612.649 2.11 0.039 1071.081 39414.73 | ||||||
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) = 12.59
Prob > F = 0.0007
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 | Coefficient Std. err. t P>|t| [95% conf. interval] | ||||||
| (1) | 7.391121 2.082689 3.55 0.001 3.237329 11.54491 | ||||||
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) Bootstrap replications (20).................... .75-.25 Interquantile regression Number of obs = 74 bootstrap(20) SEs .75 Pseudo R2 = 0.3840
1 2 3 4 5
| Bootstrap | |||||||
| price | Coefficient std. err. t P>|t| [95% conf. interval] | ||||||
| weight | 7.391121 2.082689 3.55 0.001 3.237329 11.54491 | ||||||
| length | -223.6288 74.62895 -3.00 0.004 -372.4716 -74.78609 | ||||||
| foreign | 1385.208 1420.119 0.98 0.333 -1447.13 4217.545 | ||||||
| _cons | 22122.68 9288.568 2.38 0.020 3597.215 40648.14 | ||||||
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.
Hao, Lingxin, and Daniel Q. Naiman. 2007. Quantile regression.
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.
