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 in gross domestic product (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 that are in Output 95.2.1 through Output 95.2.6.
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 uses the default simplex algorithm to estimate the parameters and uses the MCMB resampling method to compute confidence limits.
Output 95.2.1 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), indicating that this variable might have outliers.
Output 95.2.1: Model Information and Summary Statistics
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 95.2.2 displays the parameter estimates and 95% confidence limits that are computed with the rank method.
Output 95.2.2: 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, which are requested in the PLOTS= option, are displayed in Output 95.2.3 and Output 95.2.4. Output 95.2.3 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 95.2.4 plots the robust MCD distance against the Mahalanobis distance. This display is used to diagnose leverage points.
The cutoff value 8, which is specified in 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 95.2.5. This output shows that median regression fits the data well.
Output 95.2.3: Plot of Residual versus Robust Distance
Output 95.2.4: Plot of Robust Distance versus Mahalanobis Distance
Output 95.2.5: Histogram for Residuals
Tests of significance for the initial percapita GDP (LGDP2) are shown in Output 95.2.6.
Output 95.2.6: Tests for Regression Coefficient
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 by using the sparsity or resampling methods. But they cannot be computed by using the rank method, because the computation would be prohibitively expensive.
A total of 14 quantile process plots are produced. Output 95.2.7 and Output 95.2.8 display two panels of eight selected process plots. The 95% confidence bands are shaded.
Output 95.2.7: Quantile Processes with 95% Confidence Bands
Output 95.2.8: 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 in percapita GDP). The following statements request a single process plot for this effect:
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 95.2.9.
Output 95.2.9: Quantile Process Plot for LGDP2
The confidence bands here are computed by using the MCMB resampling method. In contrast, Koenker and Machado (1999) used the rank method to compute confidence limits for a few selected points. Output 95.2.9 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 consumption divided by GDP (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).