Example 82.1 Simulation Study :: SAS/STAT(R) 13.1 User's Guide

Example 82.1 Simulation Study

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} 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 of the data set indicates the true quantile level of the response y conditional on $\mb {x}=(x_1,\ldots ,x_ p)$ .

Let $Q_ y(\tau |\mb {x})=\mb {x} {\mbox{\boldmath $\beta $}}(\tau )$ denote the underlying quantile regression model, where ${\mbox{\boldmath $\beta $}}(\tau )=(\beta _1(\tau ),\ldots ,\beta _ p(\tau ))’$ . Then, the true parameter functions are

$\begin{eqnarray*} \beta _1(\tau )& =& \tau -0.1\\ \beta _2(\tau )& =& \tau ^2-0.25\\ \beta _3(\tau )& =& \exp (\tau )-\exp (0.9)\\ \beta _4(\tau )& =& ...=\beta _ p(\tau )=0 \end{eqnarray*}$

It is easy to see that, at $\tau =0.1$ , only $\beta _2(0.1)=-0.24$ and $\beta _3(0.1)=\exp (0.1)-\exp (0.9)\approx -1.354432$ are nonzero parameters. Therefore, an effective effect selection method should select $x_2$ and $x_3$ and drop all the other effects in this data set at $\tau =0.1$ . By the same rationale, $x_1$ and $x_3$ should be selected at $\tau =0.5$ with $\beta _1(0.5)=0.4$ and $\beta _3(0.5)\approx -0.810882$ , and $x_1$ and $x_2$ should be selected at $\tau =0.9$ with $\beta _1(0.9)=0.8$ and $\beta _2(0.9)=0.56$ .

The following statements use PROC QUANTSELECT with the adaptive LASSO method:

proc quantselect data=analysisData;
   model y= x1-x&p / quantile=0.1 0.5 0.9
         selection=lasso(adaptive);
   output out=out p=pred;
run;

Output 82.1.1 shows that, by default, the CHOOSE= and STOP= options are both set to SBC.

Output 82.1.1: Model Information

The QUANTSELECT Procedure

Model Information
Data Set	WORK.ANALYSISDATA
Dependent Variable	y
Selection Method	Adaptive LASSO
Quantile Type	Single Level
Stop Criterion	SBC
Choose Criterion	SBC

The selected effects and the relevant estimates are shown in Output 82.1.2 for $\tau =0.1$ , Output 82.1.3 for $\tau =0.5$ , and Output 82.1.4 for $\tau =0.9$ . You can see that the adaptive LASSO method correctly selects active effects for all three quantile levels.

Output 82.1.2: Parameter Estimates at $\tau =0.1$

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 82.1.3: Parameter Estimates at $\tau =0.5$

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 82.1.4: Parameter Estimates at $\tau =0.9$

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 $\mb {x}$ . In this simulation study, the true mean model is

$\mbox{E}(y|\mb {x})=\mb {x}{\mbox{\boldmath $\beta $}}$

where

$\begin{eqnarray*} \beta _1& =& \mbox{E($U$)}-0.1=0.4\\ \beta _2& =& \mbox{E($U^2$)}-0.25\approx 0.083333\\ \beta _3& =& \mbox{E($\exp (U)$)}-\exp (0.9)\approx -0.741321\\ \beta _4& =& ...=\beta _ p=0 \end{eqnarray*}$

The following statements perform effect selection for the quantile process with the forward selection method.

proc quantselect data=analysisData;
   model y= x1-x&p / quantile=process(ntau=all)
         selection=forward;
run;

Output 82.1.5 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 82.1.6.

Output 82.1.5: Model Information

The QUANTSELECT Procedure

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 82.1.6: 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 82.1.7 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= x1-x3 / selection=forward(select=sbc stop=sbc choose=sbc);
run;

Output 82.1.7: Parameter Estimates

The GLMSELECT Procedure

Selected Model

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