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The ROBUSTREG Procedure

Example 75.3 Growth Study of De Long and Summers

Robust regression and outlier detection techniques have considerable applications to econometrics. The following 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 using OLS with the following data set growth.

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

   ... 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 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 ( Chapter 74, The REG Procedure ) for the OLS analysis:

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

The OLS analysis shown in Output 75.3.1 indicates that GAP and EQP have a significant influence on GDP at the level.

Output 75.3.1 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 with 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 75.3.2 displays model information and summary statistics for variables in the model.

Output 75.3.2 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
MAD
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 75.3.3 displays the M estimates. Besides GAP and EQP, the robust analysis also indicates that NEQ is significant. This new finding is explained by Output 75.3.4, which shows that Zambia, the 60th country in the data, is an outlier. Output 75.3.4 also identifies leverage points based on the robust MCD distances; however, there are no serious high-leverage points in this data set.

Output 75.3.3 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 75.3.4 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 75.3.5 displays robust versions of goodness-of-fit statistics for the model.

Output 75.3.5 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 75.3.6 and Output 75.3.7 are for outlier and leverage-point diagnostics. Output 75.3.8 and Output 75.3.9 are a histogram and a Q-Q plot of the standardized robust residuals, respectively.

Output 75.3.6 RDPLOT for growth Data
RDPLOT for growth Data

Output 75.3.7 DDPLOT for growth Data
DDPLOT for growth Data

Output 75.3.8 Histogram
Histogram

Output 75.3.9 Q-Q Plot
Q-Q Plot

The following statements invoke the ROBUSTREG procedure with 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 75.3.10 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 75.3.10 displays the LTS estimates and the LTS R Square.

Output 75.3.11 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 75.3.11 displays outlier and leverage-point diagnostics based on the LTS estimates and the robust MCD distances.

Output 75.3.12 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          

Output 75.3.12 displays the final weighted least squares estimates, which are identical to those reported in Zaman, Rousseeuw, and Orhan (2001).

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