This simulation study exemplifies the unity of motive and effect for the PROC QUANTSELECT procedure. The following statements generate a data set that is based on a naive instrumental model (Chernozhukov and Hansen, 2008):
%let seed=321; %let p=20; %let n=3000; data analysisData; array x{&p} x1x&p; do i=1 to &n; U = ranuni(&seed); x1 = ranuni(&seed); x2 = ranexp(&seed); x3 = abs(rannor(&seed)); y = x1*(U0.1) + x2*(U*U0.25) + x3*(exp(U)exp(0.9)); do j=4 to &p; x{j} = ranuni(&seed); end; output; end; run;
Variable U
of the data set indicates the true quantile level of the response y conditional on .
Let denote the underlying quantile regression model, where . Then, the true parameter functions are












It is easy to see that, at , only and are nonzero parameters. Therefore, an effective effect selection method should select and and drop all the other effects in this data set at . By the same rationale, and should be selected at with and , and and should be selected at with and .
The following statements use PROC QUANTSELECT with the adaptive LASSO method:
proc quantselect data=analysisData; model y= x1x&p / quantile=0.1 0.5 0.9 selection=lasso(adaptive); run;
The selected effects and the relevant estimates are shown in Output 78.1.1 for , Output 78.1.2 for , and Output 78.1.3 for . You can see that the adaptive LASSO method correctly selects active effects for all three quantile levels.
Output 78.1.1: Parameter Estimates at
Selected Effects:  Intercept x2 x3 

Parameter Estimates  

Parameter  DF  Estimate  Standardized Estimate 
Intercept  1  0.011793  0 
x2  1  0.228709  0.218287 
x3  1  1.379907  0.784520 
Output 78.1.2: Parameter Estimates at
Selected Effects:  Intercept x1 x3 

Parameter Estimates  

Parameter  DF  Estimate  Standardized Estimate 
Intercept  1  0.011778  0 
x1  1  0.425843  0.118792 
x3  1  0.863316  0.490822 
Output 78.1.3: Parameter Estimates at
Selected Effects:  Intercept x1 x2 

Parameter Estimates  

Parameter  DF  Estimate  Standardized Estimate 
Intercept  1  0.007738  0 
x1  1  0.782942  0.218407 
x2  1  0.576445  0.550177 
The QUANTSELECT procedure can perform effect selection not only at a single quantile level but also for the entire quantile process. You can specify the QUANTILE=PROCESS option to do effect selection for the entire quantile process. With the QUANTILE=PROCESS option specified, the ParameterEstimates table produced by the QUANTSELECT procedure actually shows the mean prediction model of y conditional on . In this simulation study, the true mean model is

where












The following statements perform effect selection for the quantile process with the forward selection method.
proc quantselect data=analysisData; model y= x1x&p / quantile=process(ntau=all) selection=forward; run;
Output 78.1.4 shows that, by default, the SELECT= and STOP= options are both set to SBC. The selected effects and the relevant estimates for the conditional mean model are shown in Output 78.1.5.
Output 78.1.4: Model Information
Model Information  

Data Set  WORK.ANALYSISDATA 
Dependent Variable  y 
Selection Method  Forward 
Quantile Type  Process 
Select Criterion  SBC 
Stop Criterion  SBC 
Choose Criterion  SBC 
Output 78.1.5: Parameter Estimates
Parameter Estimates  

Parameter  DF  Estimate  Standardized Estimate 
Intercept  1  0.007833  0 
x1  1  0.418825  0.116834 
x2  1  0.094791  0.090472 
x3  1  0.785686  0.446687 
Linear regression is the most popular method for estimating conditional means. The following statements show how to select effects with the GLMSELECT procedure, and Output 78.1.6 shows the resulting selected effects and their estimates. You can see that the mean estimates from the QUANTSELECT procedure are similar to those from the GLMSELECT procedure. However, quantile regression can provide detailed distribution information, which is not available from linear regression.
proc glmselect data=analysisData; model y= x1x3 / selection=forward(select=sbc stop=sbc choose=sbc); run;
Output 78.1.6: Parameter Estimates
Parameter Estimates  

Parameter  DF  Estimate  Standard Error  t Value 
Intercept  1  0.010143  0.043129  0.24 
x1  1  0.434553  0.057385  7.57 
x2  1  0.114183  0.016771  6.81 
x3  1  0.797194  0.028156  28.31 