This example illustrates how to use SMM to estimate an AR(1) regression model for the following process:
In the following SAS statements, is simulated by using this model, and the endogenous variable is set to be equal to . The MOMENT statement creates two more moments for the estimation. One is the second moment, and the other is the first-order autocovariance. The NPREOBS=10 option instructs PROC MODEL to run the simulation 10 times before is compared to the first observation of . Because the initial is zero, the first is . Without the NPREOBS option, this is matched with the first observation of . With NPREOBS, this and the next nine are thrown away, and the moment match starts with the eleventh with the first observation of . This way, the initial values do not exert a large influence on the simulated endogenous variables.
%let nobs=500; data ardata; lu =0; do i=-10 to &nobs; x = rannor( 1011 ); e = rannor( 1011 ); u = .6 * lu + 1.5 * e; Y = 2 + 1.5 * x + u; lu = u; if i > 0 then output; end; run; title1 'Simulated Method of Moments for AR(1) Process'; proc model data=ardata ; parms a b s 1 alpha .5; instrument x; u = alpha * zlag(u) + s * rannor( 8003 ); ysim = a + b * x + u; y = ysim; moment y = (2) lag1(1); fit y / gmm npreobs=10 ndraw=10; bound s > 0, 1 > alpha > 0; run;
The output of the MODEL procedure is shown in Output 19.16.1:
Simulated Method of Moments for AR(1) Process |
Model Summary | |
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Model Variables | 1 |
Parameters | 4 |
Equations | 3 |
Number of Statements | 8 |
Program Lag Length | 1 |
Model Variables | Y |
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Parameters(Value) | a b s(1) alpha(0.5) |
Equations | _moment_2 _moment_1 Y |
The 3 Equations to Estimate | |
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_moment_2 = | F(a, b, s, alpha) |
_moment_1 = | F(a, b, s, alpha) |
Y = | F(a(1), b(x), s, alpha) |
Instruments | 1 x |
Nonlinear GMM Parameter Estimates | ||||
---|---|---|---|---|
Parameter | Estimate | Approx Std Err | t Value | Approx Pr > |t| |
a | 1.632798 | 0.1038 | 15.73 | <.0001 |
b | 1.513197 | 0.0698 | 21.67 | <.0001 |
s | 1.427888 | 0.0984 | 14.52 | <.0001 |
alpha | 0.543985 | 0.0809 | 6.72 | <.0001 |