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 .

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= x1-x&p / quantile=0.1 0.5 0.9 selection=lasso(adaptive); output out=out p=pred; run;

Output 96.1.1 shows that, by default, the CHOOSE= and STOP= options are both set to SBC.

Output 96.1.1: Model Information

The selected effects and the relevant estimates are shown in Output 96.1.2 for , Output 96.1.3 for , and Output 96.1.4 for . You can see that the adaptive LASSO method correctly selects active effects for all three quantile levels.

Output 96.1.2: Parameter Estimates at

Output 96.1.3: Parameter Estimates at

Output 96.1.4: Parameter Estimates at

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= x1-x&p / quantile=process(ntau=all) selection=forward; run;

Output 96.1.5 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 96.1.6.

Output 96.1.5: Model Information

Output 96.1.6: Parameter Estimates

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 96.1.7 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 96.1.7: Parameter Estimates