# 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
Number of Iterations 17
Optimization Method Quasi-Newton
AIC 704.21444
Schwarz Criterion 735.71473

Parameter Estimates
Parameter DF Estimate Standard
Error
tÂ 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 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
Number of Iterations 19
Optimization Method Quasi-Newton
AIC 815.57015
Schwarz Criterion 849.28702

Parameter Estimates
Parameter DF Estimate Standard
Error
tÂ 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