# The HPQUANTSELECT Procedure

### Example 13.1 Simulation Study

This example is based on Simulation Study in SAS/STAT 13.2 User's Guide. This simulation study shows how you can use the forward selection method to select quantile regression models for single quantile levels. The following statements simulate a data set from a naive instrumental model (Chernozhukov and Hansen, 2008):

%let seed=321;
%let p=20;
%let n=3000;

data analysisData;
array x{&p} x1-x&p;
do i=1 to &n;
U  = ranuni(&seed);
x1 = ranuni(&seed);
x2 = ranexp(&seed);
x3 = abs(rannor(&seed));
y  = x1*(U-0.1) + x2*(U*U-0.25) + x3*(exp(U)-exp(0.9));
do j=4 to &p;
x{j} = ranuni(&seed);
end;
output;
end;
run;


Variable U in 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 x2 and x3 and drop all the other effects in this data set at . By the same rationale, x1 and x3 should be selected at with and , and x1 and x2 should be selected at with and .

The following statements use PROC HPQUANTSELECT with the forward selection method. The STB option and the CLB option in the MODEL statement request the standardized parameter estimates and the confidence limits of parameter estimates, respectively.

proc hpquantselect data=analysisData;
model y= x1-x&p / quantile=0.1 0.5 0.9 stb clb;
selection method=forward;
output out=out p=pred;
run;


Output 13.1.1 shows that, by default, the CHOOSE= and STOP= options are both set to SBC.

Output 13.1.1: Model Information

The HPQUANTSELECT Procedure

Selection Information
Selection Method Forward
Select Criterion SBC
Stop Criterion SBC
Effect Hierarchy Enforced None
Stop Horizon 3

Output 13.1.2, Output 13.1.3, and Output 13.1.4 display the selected effects and the parameter estimates for , , and , respectively. You can see that the forward selection method correctly selects active effects for all three quantile levels.

Output 13.1.2: Parameter Estimates at

The HPQUANTSELECT Procedure
Quantile Level = 0.1
Selected Model

Selected Effects: Intercept x2 x3

Parameter Estimates
Parameter DF Estimate Standardized
Estimate
Standard
Error
95% Confidence Limits
Intercept 1 0.01179 0 0.01192 -0.01158 0.03516
x2 1 -0.22871 -0.21829 0.00946 -0.24725 -0.21017
x3 1 -1.37991 -0.78452 0.01556 -1.41042 -1.34939

Output 13.1.3: Parameter Estimates at

The HPQUANTSELECT Procedure
Quantile Level = 0.5
Selected Model

Selected Effects: Intercept x1 x3

Parameter Estimates
Parameter DF Estimate Standardized
Estimate
Standard
Error
95% Confidence Limits
Intercept 1 0.01178 0 0.03418 -0.05524 0.07879
x1 1 0.42584 0.11879 0.06237 0.30355 0.54814
x3 1 -0.86332 -0.49082 0.04765 -0.95674 -0.76989

Output 13.1.4: Parameter Estimates at

The HPQUANTSELECT Procedure
Quantile Level = 0.9
Selected Model

Selected Effects: Intercept x1 x2

Parameter Estimates
Parameter DF Estimate Standardized
Estimate
Standard
Error
95% Confidence Limits
Intercept 1 -0.00774 0 0.03292 -0.07228 0.05680
x1 1 0.78294 0.21841 0.05134 0.68228 0.88360
x2 1 0.57644 0.55018 0.03422 0.50935 0.64354