# The QUANTREG Procedure

### Example 77.2 Quantile Regression for Econometric Growth Data

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`

Country’s name and period

`GDP`

Annual change per capita GDP

`lgdp2`

Initial per capita GDP

`mse2`

Male secondary education

`fse2`

Female secondary education

`fhe2`

Female higher education

`mhe2`

Male higher education

`lexp2`

Life expectancy

`lintr2`

Human capital

`gedy2`

EducationGDP

`Iy2`

InvestmentGDP

`gcony2`

Public consumptionGDP

`lblakp2`

`pol2`

Political instability

`ttrad2`

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: 2-80 Years Old

The QUANTREG Procedure

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
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: Residual-Robust Distance Plot

Output 77.2.4: Robust Distance-Mahalanobis Distance Plot

Output 77.2.5: Histogram for Residuals

Tests of significance for the initial per-capita 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 Chi-Square 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 per-capita GDP on the growth of this variable (annual change per-capita 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).