# The HPQUANTSELECT Procedure

### Example 59.1 Simulation Study

This example is based on Simulation Study. 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 59.1.1 shows that, by default, the CHOOSE= and STOP= options are both set to SBC.

Output 59.1.1: Model Information

The HPQUANTSELECT Procedure

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

Output 59.1.2, Output 59.1.3, and Output 59.1.4 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 59.1.2: 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 t Value Pr > |t|
Intercept 1 0.01179 0 0.01192 -0.01158 0.03516 0.99 0.3225
x2 1 -0.22871 -0.21829 0.00946 -0.24725 -0.21017 -24.19 <.0001
x3 1 -1.37991 -0.78452 0.01556 -1.41042 -1.34939 -88.67 <.0001

Output 59.1.3: 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 t Value Pr > |t|
Intercept 1 0.01178 0 0.03418 -0.05524 0.07879 0.34 0.7304
x1 1 0.42584 0.11879 0.06237 0.30355 0.54814 6.83 <.0001
x3 1 -0.86332 -0.49082 0.04765 -0.95674 -0.76989 -18.12 <.0001

Output 59.1.4: 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 t Value Pr > |t|
Intercept 1 -0.00774 0 0.03292 -0.07228 0.05680 -0.24 0.8142
x1 1 0.78294 0.21841 0.05134 0.68228 0.88360 15.25 <.0001
x2 1 0.57644 0.55018 0.03422 0.50935 0.64354 16.85 <.0001