The QLIM Procedure

Example 22.5 Sample Selection Model with Truncation and Censoring

In this example the data are generated such that the selection variable is discrete and the variable Y is truncated from below by zero. The program follows, with the results shown in Output 22.5.1:

data trunc;
   keep z y x1 x2;
   do i = 1 to 500;
      x1 = rannor( 19283 );
      x2 = rannor( 19283 );
      u1 = rannor( 19283 );
      u2 = rannor( 19283 );
      zl = 1 + 2 * x1 + 3 * x2 + u1;
      y = 3 + 4 * x1 - 2 * x2 + u1*.2 + u2;
      if ( zl > 0 ) then z = 1;
      else z = 0;
      if y>=0 then output;
   end;
run;

 /*-- Sample Selection with Truncation --*/
 proc qlim data=trunc method=qn;
    model z = x1 x2 / discrete;
    model y = x1 x2 / select(z=1) truncated(lb=0);
 run;

Output 22.5.1 Sample Selection with Truncation

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	2
Endogenous Variable	z y
Number of Observations	379
Log Likelihood	-344.10722
Maximum Absolute Gradient	4.95535E-6
Number of Iterations	17
Optimization Method	Quasi-Newton
AIC	704.21444
Schwarz Criterion	735.71473

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
y.Intercept	1	3.014158	0.128548	23.45	<.0001
y.x1	1	3.995671	0.099599	40.12	<.0001
y.x2	1	-1.972697	0.096385	-20.47	<.0001
_Sigma.y	1	0.923428	0.047233	19.55	<.0001
z.Intercept	1	0.949444	0.190265	4.99	<.0001
z.x1	1	2.163928	0.288384	7.50	<.0001
z.x2	1	3.134213	0.379251	8.26	<.0001
_Rho	1	0.494356	0.176542	2.80	0.0051

In the following statements the data are generated such that the selection variable is discrete and the variable $\text{[math]}$ is censored from below by zero. The results are shown in Output 22.5.2.

data cens;
   keep z y x1 x2;
   do i = 1 to 500;
      x1 = rannor( 19283 );
      x2 = rannor( 19283 );
      u1 = rannor( 19283 );
      u2 = rannor( 19283 );
      zl = 1 + 2 * x1 + 3 * x2 + u1;
      yl = 3 + 4 * x1 - 2 * x2 + u1*.2 + u2;
      if ( zl > 0 ) then z = 1;
      else               z = 0;
      if ( yl > 0 ) then y = yl;
      else               y = 0;
      output;
   end;
run;

 /*-- Sample Selection with Censoring --*/
 proc qlim data=cens method=qn;
    model z = x1 x2 / discrete;
    model y = x1 x2 / select(z=1) censored(lb=0);
 run;

Output 22.5.2 Sample Selection with Censoring

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	2
Endogenous Variable	z y
Number of Observations	500
Log Likelihood	-399.78508
Maximum Absolute Gradient	2.30443E-6
Number of Iterations	19
Optimization Method	Quasi-Newton
AIC	815.57015
Schwarz Criterion	849.28702

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
y.Intercept	1	3.074276	0.111617	27.54	<.0001
y.x1	1	3.963619	0.085796	46.20	<.0001
y.x2	1	-2.023548	0.088714	-22.81	<.0001
_Sigma.y	1	0.920860	0.043278	21.28	<.0001
z.Intercept	1	1.013610	0.154081	6.58	<.0001
z.x1	1	2.256922	0.255999	8.82	<.0001
z.x2	1	3.302692	0.342168	9.65	<.0001
_Rho	1	0.350776	0.197093	1.78	0.0751