The QLIM Procedure

Example 22.6 Types of Tobit Models

The following five examples show how to estimate different types of Tobit models (see Types of Tobit Models). Output 22.6.1 through Output 22.6.5 show the results of the corresponding programs.

Type 1 Tobit

data a1;
   keep y x;
   do i = 1 to 500;
      x = rannor( 19283 );
      u = rannor( 19283 );
      yl = 1 + 2 * x + u;
      if ( yl > 0 ) then y = yl;
      else                y = 0;
      output;
   end;
run;

/*-- Type 1 Tobit --*/
proc qlim data=a1 method=qn;
   model y = x;
   endogenous y ~ censored(lb=0);
run;

Output 22.6.1: Type 1 Tobit

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	1
Endogenous Variable	y
Number of Observations	500
Log Likelihood	-554.17696
Maximum Absolute Gradient	4.65556E-7
Number of Iterations	9
Optimization Method	Quasi-Newton
AIC	1114
Schwarz Criterion	1127

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Intercept	1	0.942734	0.056784	16.60	<.0001
x	1	2.049571	0.066979	30.60	<.0001
_Sigma	1	1.016571	0.039035	26.04	<.0001

Type 2 Tobit

data a2;
   keep y1 y2 x1 x2;
   do i = 1 to 500;
      x1 = rannor( 19283 );
      x2 = rannor( 19283 );
      u1 = rannor( 19283 );
      u2 = rannor( 19283 );
      y1l = 1 + 2 * x1 + 3 * x2 + u1;
      y2l = 3 + 4 * x1 - 2 * x2 + u1*.2 + u2;
      if ( y1l > 0 ) then y1 = 1;
      else                y1 = 0;
      if ( y1l > 0 ) then y2 = y2l;
      else                y2 = 0;
      output;
   end;
run;

/*-- Type 2 Tobit --*/
proc qlim data=a2 method=qn;
   model y1 = x1 x2 / discrete;
   model y2 = x1 x2 / select(y1=1);
run;

Output 22.6.2: Type 2 Tobit

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	2
Endogenous Variable	y1 y2
Number of Observations	500
Log Likelihood	-476.12328
Maximum Absolute Gradient	8.30075E-7
Number of Iterations	17
Optimization Method	Quasi-Newton
AIC	968.24655
Schwarz Criterion	1002

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
y2.Intercept	1	3.066992	0.106903	28.69	<.0001
y2.x1	1	4.004874	0.072043	55.59	<.0001
y2.x2	1	-2.079352	0.087544	-23.75	<.0001
_Sigma.y2	1	0.940559	0.039321	23.92	<.0001
y1.Intercept	1	1.017140	0.154975	6.56	<.0001
y1.x1	1	2.253080	0.256097	8.80	<.0001
y1.x2	1	3.305140	0.343695	9.62	<.0001
_Rho	1	0.292992	0.210073	1.39	0.1631

Type 3 Tobit

data a3;
   keep y1 y2 x1 x2;
   do i = 1 to 500;
      x1 = rannor( 19283 );
      x2 = rannor( 19283 );
      u1 = rannor( 19283 );
      u2 = rannor( 19283 );
      y1l = 1 + 2 * x1 + 3 * x2 + u1;
      y2l = 3 + 4 * x1 - 2 * x2 + u1*.2 + u2;
      if ( y1l > 0 ) then y1 = y1l;
      else                y1 = 0;
      if ( y1l > 0 ) then y2 = y2l;
      else                y2 = 0;
      output;
   end;
run;

/*-- Type 3 Tobit --*/
proc qlim data=a3 method=qn;
   model y1 = x1 x2 / censored(lb=0);
   model y2 = x1 x2 / select(y1>0);
run;

Output 22.6.3: Type 3 Tobit

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	2
Endogenous Variable	y1 y2
Number of Observations	500
Log Likelihood	-838.94087
Maximum Absolute Gradient	9.71691E-6
Number of Iterations	16
Optimization Method	Quasi-Newton
AIC	1696
Schwarz Criterion	1734

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
y2.Intercept	1	3.081206	0.080121	38.46	<.0001
y2.x1	1	3.998361	0.063734	62.73	<.0001
y2.x2	1	-2.088280	0.072876	-28.66	<.0001
_Sigma.y2	1	0.939799	0.039047	24.07	<.0001
y1.Intercept	1	0.981975	0.067351	14.58	<.0001
y1.x1	1	2.032675	0.059363	34.24	<.0001
y1.x2	1	2.976609	0.065584	45.39	<.0001
_Sigma.y1	1	0.969968	0.039795	24.37	<.0001
_Rho	1	0.226281	0.057672	3.92	<.0001

Type 4 Tobit

data a4;
   keep y1 y2 y3 x1 x2;
   do i = 1 to 500;
      x1 = rannor( 19283 );
      x2 = rannor( 19283 );
      u1 = rannor( 19283 );
      u2 = rannor( 19283 );
      u3 = rannor( 19283 );
      y1l = 1 + 2 * x1 + 3 * x2 + u1;
      y2l = 3 + 4 * x1 - 2 * x2 + u1*.2 + u2;
      y3l = 0 - 1 * x1 + 1 * x2 + u1*.1 - u2*.5 + u3*.5;
      if ( y1l > 0 ) then y1 = y1l;
      else                y1 = 0;
      if ( y1l > 0 ) then y2 = y2l;
      else                y2 = 0;
      if ( y1l <= 0 ) then y3 = y3l;
      else                y3 = 0;
      output;
   end;
run;

/*-- Type 4 Tobit --*/
proc qlim data=a4 method=qn;
   model y1 = x1 x2 / censored(lb=0);
   model y2 = x1 x2 / select(y1>0);
   model y3 = x1 x2 / select(y1<=0);
run;

Output 22.6.4: Type 4 Tobit

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	3
Endogenous Variable	y1 y2 y3
Number of Observations	500
Log Likelihood	-1128
Maximum Absolute Gradient	0.0000161
Number of Iterations	21
Optimization Method	Quasi-Newton
AIC	2285
Schwarz Criterion	2344

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
y2.Intercept	1	2.894656	0.076079	38.05	<.0001
y2.x1	1	4.072704	0.062675	64.98	<.0001
y2.x2	1	-1.901163	0.076874	-24.73	<.0001
_Sigma.y2	1	0.981655	0.039564	24.81	<.0001
y3.Intercept	1	0.064594	0.179441	0.36	0.7189
y3.x1	1	-0.938384	0.096570	-9.72	<.0001
y3.x2	1	1.035798	0.123104	8.41	<.0001
_Sigma.y3	1	0.743124	0.038240	19.43	<.0001
y1.Intercept	1	0.987370	0.067861	14.55	<.0001
y1.x1	1	2.050408	0.060819	33.71	<.0001
y1.x2	1	2.982190	0.072552	41.10	<.0001
_Sigma.y1	1	1.032473	0.040971	25.20	<.0001
_Rho.y1.y2	1	0.291587	0.053436	5.46	<.0001
_Rho.y1.y3	1	-0.031665	0.260057	-0.12	0.9031

Type 5 Tobit

data a5;
   keep y1 y2 y3 x1 x2;
   do i = 1 to 500;
      x1 = rannor( 19283 );
      x2 = rannor( 19283 );
      u1 = rannor( 19283 );
      u2 = rannor( 19283 );
      u3 = rannor( 19283 );
      y1l = 1 + 2 * x1 + 3 * x2 + u1;
      y2l = 3 + 4 * x1 - 2 * x2 + u1*.2 + u2;
      y3l = 0 - 1 * x1 + 1 * x2 + u1*.1 - u2*.5 + u3*.5;
      if ( y1l > 0 ) then y1 = 1;
      else                y1 = 0;
      if ( y1l > 0 ) then y2 = y2l;
      else                y2 = 0;
      if ( y1l <= 0 ) then y3 = y3l;
      else                y3 = 0;
      output;
   end;
run;

/*-- Type 5 Tobit --*/
proc qlim data=a5 method=qn;
   model y1 = x1 x2 / discrete;
   model y2 = x1 x2 / select(y1>0);
   model y3 = x1 x2 / select(y1<=0);
run;

Output 22.6.5: Type 5 Tobit

Binary Data

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables	3
Endogenous Variable	y1 y2 y3
Number of Observations	500
Log Likelihood	-734.50612
Maximum Absolute Gradient	3.57134E-7
Number of Iterations	20
Optimization Method	Quasi-Newton
AIC	1495
Schwarz Criterion	1550

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
y2.Intercept	1	2.887523	0.095193	30.33	<.0001
y2.x1	1	4.078926	0.069623	58.59	<.0001
y2.x2	1	-1.898898	0.086578	-21.93	<.0001
_Sigma.y2	1	0.983059	0.039987	24.58	<.0001
y3.Intercept	1	0.071764	0.171522	0.42	0.6757
y3.x1	1	-0.935299	0.092843	-10.07	<.0001
y3.x2	1	1.039954	0.120697	8.62	<.0001
_Sigma.y3	1	0.743083	0.038225	19.44	<.0001
y1.Intercept	1	1.067578	0.142789	7.48	<.0001
y1.x1	1	2.068376	0.226020	9.15	<.0001
y1.x2	1	3.157385	0.314743	10.03	<.0001
_Rho.y1.y2	1	0.312369	0.177010	1.76	0.0776
_Rho.y1.y3	1	-0.018225	0.234886	-0.08	0.9382