# The ROBUSTREG Procedure

### Example 86.3 Growth Study of De Long and Summers

Robust regression and outlier detection techniques have considerable applications to econometrics. This example, from Zaman, Rousseeuw, and Orhan (2001), shows how these techniques substantially improve the ordinary least squares (OLS) results for the growth study of De Long and Summers.

De Long and Summers (1991) studied the national growth of 61 countries from 1960 to 1985 by applying OLS to the following data set:

data growth;
input country \$ GDP LFG EQP NEQ GAP @@;
datalines;
Argentin  0.0089 0.0118 0.0214 0.2286 0.6079
Austria   0.0332 0.0014 0.0991 0.1349 0.5809
Belgium   0.0256 0.0061 0.0684 0.1653 0.4109
Bolivia   0.0124 0.0209 0.0167 0.1133 0.8634
Botswana  0.0676 0.0239 0.1310 0.1490 0.9474

... more lines ...

Venezuel  0.0120 0.0378 0.0340 0.0760 0.4974
Zambia   -0.0110 0.0275 0.0702 0.2012 0.8695
Zimbabwe  0.0110 0.0309 0.0843 0.1257 0.8875
;


The regression equation that they used is

where the response variable is the growth in gross domestic product per worker (GDP) and the regressors are labor force growth (LFG), relative GDP gap (GAP), equipment investment (EQP), and nonequipment investment (NEQ).

The following statements invoke the REG procedure (see Chapter 85: The REG Procedure) for the OLS analysis:

proc reg data=growth;
model GDP  = LFG GAP EQP NEQ;
run;


The OLS analysis that is shown in Output 86.3.1 indicates that GAP and EQP have a significant influence on GDP at the 5% level.

Output 86.3.1: OLS Estimates

The REG Procedure
Model: MODEL1
Dependent Variable: GDP

Parameter Estimates
Variable DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept 1 -0.01430 0.01028 -1.39 0.1697
LFG 1 -0.02981 0.19838 -0.15 0.8811
GAP 1 0.02026 0.00917 2.21 0.0313
EQP 1 0.26538 0.06529 4.06 0.0002
NEQ 1 0.06236 0.03482 1.79 0.0787

The following statements invoke the ROBUSTREG procedure and use the default M estimation:

ods graphics on;

proc robustreg data=growth plots=all;
model GDP  = LFG GAP EQP NEQ / diagnostics leverage;
id country;
run;

ods graphics off;


Output 86.3.2 displays model information and summary statistics for variables in the model.

Output 86.3.2: Model-Fitting Information and Summary Statistics

The ROBUSTREG Procedure

Model Information
Data Set WORK.GROWTH
Dependent Variable GDP
Number of Independent Variables 4
Number of Observations 61
Method M Estimation

Summary Statistics
Variable Q1 Median Q3 Mean Standard
Deviation
LFG 0.0118 0.0239 0.0281 0.0211 0.00979 0.00949
GAP 0.5796 0.8015 0.8863 0.7258 0.2181 0.1778
EQP 0.0265 0.0433 0.0720 0.0523 0.0296 0.0325
NEQ 0.0956 0.1356 0.1812 0.1399 0.0570 0.0624
GDP 0.0121 0.0231 0.0310 0.0224 0.0155 0.0150

Output 86.3.3 displays the M estimates. Besides GAP and EQP, the robust analysis also indicates that NEQ is significant. This new finding is explained by Output 86.3.4, which shows that Zambia, the 60th country in the data, is an outlier. Output 86.3.4 also identifies leverage points that are based on the robust MCD distances; however, there are no serious high-leverage points in this data set.

Output 86.3.3: M Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -0.0247 0.0097 -0.0437 -0.0058 6.53 0.0106
LFG 1 0.1040 0.1867 -0.2619 0.4699 0.31 0.5775
GAP 1 0.0250 0.0086 0.0080 0.0419 8.36 0.0038
EQP 1 0.2968 0.0614 0.1764 0.4172 23.33 <.0001
NEQ 1 0.0885 0.0328 0.0242 0.1527 7.29 0.0069
Scale 1 0.0099

Output 86.3.4: Diagnostics

Diagnostics
Obs country Mahalanobis Distance Robust MCD Distance Leverage Standardized
Robust Residual
Outlier
1 Argentin 2.6083 4.0639 * -0.9424
5 Botswana 3.4351 6.7391 * 1.4200
8 Canada 3.1876 4.6843 * -0.1972
9 Chile 3.6752 5.0599 * -1.8784
17 Finland 2.6024 3.8186 * -1.7971
23 HongKong 2.1225 3.8238 * 1.7161
27 Israel 2.6461 5.0336 * 0.0909
31 Japan 2.9179 4.7140 * 0.0216
53 Tanzania 2.2600 4.3193 * -1.8082
57 U.S. 3.8701 5.4874 * 0.1448
58 Uruguay 2.5953 3.9671 * -0.0978
59 Venezuel 2.9239 4.1663 * 0.3573
60 Zambia 1.8562 2.7135   -4.9798 *
61 Zimbabwe 1.9634 3.9128 * -2.5959

Output 86.3.5 displays robust versions of goodness-of-fit statistics for the model.

Output 86.3.5: Goodness-of-Fit Statistics

Goodness-of-Fit
Statistic Value
R-Square 0.3178
AICR 80.2134
BICR 91.5095
Deviance 0.0070

The PLOTS=ALL option generates four diagnostic plots. Output 86.3.6 and Output 86.3.7 are for outlier and leverage-point diagnostics. Output 86.3.8 and Output 86.3.9 are a histogram and a Q-Q plot, respectively, of the standardized robust residuals.

Output 86.3.6: RD Plot for Growth Data

Output 86.3.7: DD Plot for Growth Data

Output 86.3.8: Histogram

Output 86.3.9: Q-Q Plot

The following statements invoke the ROBUSTREG procedure and use LTS estimation, which was used by Zaman, Rousseeuw, and Orhan (2001). The results are consistent with those of M estimation.

proc robustreg method=lts(h=33) fwls data=growth seed=100;
model GDP  = LFG GAP EQP NEQ / diagnostics leverage;
id country;
run;


Output 86.3.10 displays the LTS estimates and the LTS R square.

Output 86.3.10: LTS Estimates and LTS R Square

The ROBUSTREG Procedure

LTS Parameter Estimates
Parameter DF Estimate
Intercept 1 -0.0249
LFG 1 0.1123
GAP 1 0.0214
EQP 1 0.2669
NEQ 1 0.1110
Scale (sLTS) 0 0.0076
Scale (Wscale) 0 0.0109

R-Square for LTS Estimation
R-Square 0.7418

Output 86.3.11 displays outlier and leverage-point diagnostics that are based on the LTS estimates and the robust MCD distances.

Output 86.3.11: Diagnostics

Diagnostics
Obs country Mahalanobis Distance Robust MCD Distance Leverage Standardized
Robust Residual
Outlier
1 Argentin 2.6083 4.0639 * -1.0715
5 Botswana 3.4351 6.7391 * 1.6574
8 Canada 3.1876 4.6843 * -0.2324
9 Chile 3.6752 5.0599 * -2.0896
17 Finland 2.6024 3.8186 * -1.6367
23 HongKong 2.1225 3.8238 * 1.7570
27 Israel 2.6461 5.0336 * 0.2334
31 Japan 2.9179 4.7140 * 0.0971
53 Tanzania 2.2600 4.3193 * -1.2978
57 U.S. 3.8701 5.4874 * 0.0605
58 Uruguay 2.5953 3.9671 * -0.0857
59 Venezuel 2.9239 4.1663 * 0.4113
60 Zambia 1.8562 2.7135   -4.4984 *
61 Zimbabwe 1.9634 3.9128 * -2.1201

Output 86.3.12 displays the final weighted least squares estimates, which are identical to those that are reported in Zaman, Rousseeuw, and Orhan (2001).

Output 86.3.12: Final Weighted LS Estimates

Parameter Estimates for Final Weighted Least Squares Fit
Parameter DF Estimate Standard
Error
95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -0.0222 0.0093 -0.0405 -0.0039 5.65 0.0175
LFG 1 0.0446 0.1771 -0.3026 0.3917 0.06 0.8013
GAP 1 0.0245 0.0082 0.0084 0.0406 8.89 0.0029
EQP 1 0.2824 0.0581 0.1685 0.3964 23.60 <.0001
NEQ 1 0.0849 0.0314 0.0233 0.1465 7.30 0.0069
Scale 0 0.0116