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):
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.
SMM for Duration Model with Unobserved Heterogeneity |
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 |