Example 19.18 Duration Data Model with Unobserved Heterogeneity

All of the previous three models actually have closed-form moment conditions, so the simulation approach is not necessarily required for the estimation. This example illustrates how to use SMM to estimate a model for which there is no closed-form solution for the moments and thus the traditional GMM method does not apply. The model is the duration data model with unobserved heterogeneity in Gourieroux and Monfort (1993):

$\displaystyle y_ i$	$\displaystyle =$	$\displaystyle - exp( - b x_ i - \sigma u_ i ) log(v_ i)$
$\displaystyle u_ i$	$\displaystyle \sim$	$\displaystyle N(0, 1) ~ ~ ~ v_ i \sim U_{[0,1]} \nonumber$

The SAS statements are:


title1 'SMM for Duration Model with Unobserved Heterogeneity';

%let nobs=1000;
data durationdata;
   b=0.9; s=0.5;
   do i=1 to &nobs;
      u = rannor( 1011 );
      v = ranuni( 1011 );
      x = 2 * ranuni( 1011 );
      y = -exp(-b * x + s * u) * log(v);
      output;
   end;
run;

proc model data=durationdata;
   parms b .5 s 1;
   instrument x;

   u = rannor( 1011 );
   v = ranuni( 1011 );
   y = -exp(-b * x + s * u) * log(v);

   moment y = (2 3 4);
   fit y / gmm ndraw=10 ;* maxiter=500;
   bound s > 0, b > 0;
run;

The output of the MODEL procedure is shown in Output 19.18.1.

Output 19.18.1: PROC MODEL Output

SMM for Duration Model with Unobserved Heterogeneity

The MODEL Procedure

Model Summary
Model Variables	1
Parameters	2
Equations	4
Number of Statements	9

Model Variables	y
Parameters(Value)	b(0.5) s(1)
Equations	_moment_3 _moment_2 _moment_1 y

The 4 Equations to Estimate
_moment_3 =	F(b, s)
_moment_2 =	F(b, s)
_moment_1 =	F(b, s)
y =	F(b, s)
Instruments	1 x

Nonlinear GMM Parameter Estimates
Parameter	Estimate	Approx Std Err	t Value	Approx Pr > \|t\|
b	0.92983	0.0331	28.08	<.0001
s	0.341825	0.0608	5.62	<.0001