This example uses a SAS data set named Growth
, which contains economic growth rates for countries during two time periods, 1965–1975 and 1975–1985. The data come from
a study by Barro and Lee (1994) and have also been analyzed by Koenker and Machado (1999).
There are 161 observations and 15 variables in the data set. The variables, which are listed in the following table, include
the national growth rates (GDP
) for the two periods, 13 covariates, and a name variable (Country
) for identifying the countries in one of the two periods.
Variable 
Description 



Country’s name and period 


Annual change per capita GDP 


Initial per capita GDP 


Male secondary education 


Female secondary education 


Female higher education 


Male higher education 


Life expectancy 


Human capital 


EducationGDP 


InvestmentGDP 


Public consumptionGDP 


Black market premium 


Political instability 


Growth rate terms trade 
The goal is to study the effect of the covariates on GDP
. The following statements request median regression for a preliminary exploration. They produce the results in Output 77.2.1 through Output 77.2.6.
data growth; length Country$ 22; input Country GDP lgdp2 mse2 fse2 fhe2 mhe2 lexp2 lintr2 gedy2 Iy2 gcony2 lblakp2 pol2 ttrad2 @@; datalines; Algeria75 .0415 7.330 .1320 .0670 .0050 .0220 3.880 .1138 .0382 .1898 .0601 .3823 .0833 .1001 Algeria85 .0244 7.745 .2760 .0740 .0070 .0370 3.978 .107 .0437 .3057 .0850 .9386 .0000 .0657 Argentina75 .0187 8.220 .7850 .6200 .0740 .1660 4.181 .4060 .0221 .1505 .0596 .1924 .3575 .011 Argentina85 .014 8.407 .9360 .9020 .1320 .2030 4.211 .1914 .0243 .1467 .0314 .3085 .7010 .052 Australia75 .0259 9.101 2.541 2.353 .0880 .2070 4.263 6.937 .0348 .3272 .0257 .0000 .0080 .016 ... more lines ... Zambia75 .0120 6.989 .3760 .1190 .0130 .0420 3.757 .4388 .0339 .3688 .2513 .3945 .0000 .032 Zambia85 .046 7.109 .4200 .2740 .0110 .0270 3.854 .8812 .0477 .1632 .2637 .6467 .0000 .033 Zimbabwe75 .0320 6.860 .1450 .0170 .0080 .0450 3.833 .7156 .0337 .2276 .0246 .1997 .0000 .040 Zimbabwe85 .011 7.180 .2200 .0650 .0060 .0400 3.944 .9296 .0520 .1559 .0518 .7862 .7161 .024 ;
ods graphics on; proc quantreg data=growth ci=resampling plots=(rdplot ddplot reshistogram); model GDP = lgdp2 mse2 fse2 fhe2 mhe2 lexp2 lintr2 gedy2 Iy2 gcony2 lblakp2 pol2 ttrad2 / quantile=.5 diagnostics leverage(cutoff=8) seed=1268; id Country; test_lgdp2: test lgdp2 / lr wald; run;
The QUANTREG procedure employs the default simplex algorithm to estimate the parameters. The MCMB resampling method is used to compute confidence limits.
Output 77.2.1 displays model information and summary statistics for the variables in the model. Six summary statistics are computed, including
the median and the median absolute deviation (MAD), which are robust measures of univariate location and scale, respectively.
For the variable lintr2
(Human Capital), both the mean and standard deviation are much larger than the corresponding robust measures, median and
MAD. This indicates that this variable might have outliers.
Output 77.2.1: Model Information and Summary Statistics
BMI Percentiles for Men: 280 Years Old 
Model Information  

Data Set  WORK.GROWTH 
Dependent Variable  GDP 
Number of Independent Variables  13 
Number of Observations  161 
Optimization Algorithm  Simplex 
Method for Confidence Limits  Resampling 
Summary Statistics  

Variable  Q1  Median  Q3  Mean  Standard Deviation 
MAD 
lgdp2  6.9890  7.7450  8.6080  7.7905  0.9543  1.1579 
mse2  0.3160  0.7230  1.2675  0.9666  0.8574  0.6835 
fse2  0.1270  0.4230  0.9835  0.7117  0.8331  0.5011 
fhe2  0.0110  0.0350  0.0890  0.0792  0.1216  0.0400 
mhe2  0.0400  0.1060  0.2060  0.1584  0.1752  0.1127 
lexp2  3.8670  4.0640  4.2430  4.0440  0.2028  0.2728 
lintr2  0.00160  0.5604  1.8805  1.4625  2.5491  1.0058 
gedy2  0.0248  0.0343  0.0466  0.0360  0.0141  0.0151 
Iy2  0.1396  0.1955  0.2671  0.2010  0.0877  0.0981 
gcony2  0.0480  0.0767  0.1276  0.0914  0.0617  0.0566 
lblakp2  0  0.0696  0.2407  0.1916  0.3070  0.1032 
pol2  0  0.0500  0.2429  0.1683  0.2409  0.0741 
ttrad2  0.0240  0.0100  0.00730  0.00570  0.0375  0.0239 
GDP  0.00290  0.0196  0.0351  0.0191  0.0248  0.0237 
Output 77.2.2 displays parameter estimates and 95% confidence limits computed with the rank method.
Output 77.2.2: Parameter Estimates
Parameter Estimates  

Parameter  DF  Estimate  Standard Error  95% Confidence Limits  t Value  Pr > t  
Intercept  1  0.0488  0.0733  0.1937  0.0961  0.67  0.5065 
lgdp2  1  0.0269  0.0041  0.0350  0.0188  6.58  <.0001 
mse2  1  0.0110  0.0080  0.0048  0.0269  1.38  0.1710 
fse2  1  0.0011  0.0088  0.0185  0.0162  0.13  0.8960 
fhe2  1  0.0148  0.0321  0.0485  0.0782  0.46  0.6441 
mhe2  1  0.0043  0.0268  0.0487  0.0573  0.16  0.8735 
lexp2  1  0.0683  0.0229  0.0232  0.1135  2.99  0.0033 
lintr2  1  0.0022  0.0015  0.0052  0.0008  1.44  0.1513 
gedy2  1  0.0508  0.1654  0.3777  0.2760  0.31  0.7589 
Iy2  1  0.0723  0.0248  0.0233  0.1213  2.92  0.0041 
gcony2  1  0.0935  0.0382  0.1690  0.0181  2.45  0.0154 
lblakp2  1  0.0269  0.0084  0.0435  0.0104  3.22  0.0016 
pol2  1  0.0301  0.0093  0.0485  0.0117  3.23  0.0015 
ttrad2  1  0.1613  0.0740  0.0149  0.3076  2.18  0.0310 
Diagnostics for the median regression fit are displayed in Output 77.2.3 and Output 77.2.4, which are requested with the PLOTS= option. Output 77.2.3 plots the standardized residuals from median regression against the robust MCD distance. This display is used to diagnose both vertical outliers and horizontal leverage points. Output 77.2.4 plots the robust MCD distance against the Mahalanobis distance. This display is used to diagnose leverage points.
The cutoff value 8 specified with the LEVERAGE option is close to the maximum of the Mahalanobis distance. Eighteen points are diagnosed as high leverage points, and almost all are countries with high human capital, which is the major contributor to the high leverage as observed from the summary statistics. Four points are diagnosed as outliers by using the default cutoff value of 3. However, these are not extreme outliers.
A histogram of the standardized residuals and two fitted density curves are displayed in Output 77.2.5. This shows that median regression fits the data well.
Output 77.2.3: ResidualRobust Distance Plot
Output 77.2.4: Robust DistanceMahalanobis Distance Plot
Output 77.2.5: Histogram for Residuals
Tests of significance for the initial percapita GDP (LGDP2) are shown in Output 77.2.6.
Output 77.2.6: Tests for Regression Coefficient
Test test_lgdp2 Results  

Test  Test Statistic  DF  ChiSquare  Pr > ChiSq 
Wald  43.2684  1  43.27  <.0001 
Likelihood Ratio  36.3047  1  36.30  <.0001 
The QUANTREG procedure computes entire quantile processes for covariates when you specify QUANTILE=PROCESS in the MODEL statement, as follows:
proc quantreg data=growth ci=resampling; model GDP = lgdp2 mse2 fse2 fhe2 mhe2 lexp2 lintr2 gedy2 Iy2 gcony2 lblakp2 pol2 ttrad2 / quantile=process plot=quantplot seed=1268; run;
Confidence limits for quantile processes can be computed with the sparsity or resampling methods, but not the rank method, because the computation would be prohibitively expensive.
A total of 14 quantile process plots are produced. Output 77.2.7 and Output 77.2.8 display two panels of eight selected process plots. The 95% confidence bands are shaded.
Output 77.2.7: Quantile Processes with 95% Confidence Bands
Output 77.2.8: Quantile Processes with 95% Confidence Bands
As pointed out by Koenker and Machado (1999), previous studies of the Barro growth data have focused on the effect of the initial percapita GDP on the growth of this variable (annual change percapita GDP). A single process plot for this effect can be requested with the following statements:
proc quantreg data=growth ci=resampling; model GDP = lgdp2 mse2 fse2 fhe2 mhe2 lexp2 lintr2 gedy2 Iy2 gcony2 lblakp2 pol2 ttrad2 / quantile=process plot=quantplot(lgdp2) seed=1268; run;
The plot is shown in Output 77.2.9.
Output 77.2.9: Quantile Process Plot for LGDP2
The confidence bands here are computed with the MCMB resampling method, unlike in Koenker and Machado (1999), where the rank method was used to compute confidence limits for a few selected points. Output 77.2.9 suggests that the effect of the initial level of GDP is relatively constant over the entire distribution, with a slightly stronger effect in the upper tail.
The effects of other covariates are quite varied. An interesting covariate is public consumptionGDP (gcony2
) (first plot in second panel), which has a constant effect over the upper half of the distribution and a larger effect in
the lower tail. For an analysis of the effects of the other covariates, see Koenker and Machado (1999).