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=20 option instructs PROC MODEL to run the simulation 20 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 , along with the next 19 , is thrown away, and the moment match starts with the twenty-first with the first observation of . This way, the initial values do not exert a large inference 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;
      eq.ysq = y*y - ysim*ysim;
      eq.ylagy = y * lag(y) - ysim * lag( ysim );
   
      fit y ysq ylagy / 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: