Example 18.16 Simulated Method of Moments—AR(1) Process

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 18.16.1: