PROC MODEL: Simulated Method of Moments—Simple Linear Regression

The MODEL Procedure

Example 18.15 Simulated Method of Moments—Simple Linear Regression

This example illustrates how to use SMM to estimate a simple linear regression model for the following process:

$\text{[math]}$

In the following SAS statements, $\text{[math]}$ is simulated, and the first moment and the second moment of $\text{[math]}$ are compared with those of the observed endogenous variable $\text{[math]}$ .

   title "Simple regression model";
   
   data regdata;
      do i=1 to 500;
         x = rannor( 1013 );
         Y = 2 + 1.5 * x + 1.5 * rannor( 9871 );
         output;
      end;
   run;
   
   proc model data=regdata;
      parms a b s;
      instrument x;
   
      ysim = (a+b*x) + s * rannor( 8003 );
      y = ysim;
      eq.ysq = y*y - ysim*ysim;
   
      fit y ysq / gmm ndraw;
      bound s > 0;
   run;

The output of the MODEL procedure is shown in Output 18.15.1:

Output 18.15.1 PROC MODEL Output

Simple regression model

The MODEL Procedure

Model Summary
Model Variables	1
Parameters	3
Equations	2
Number of Statements	4

Model Variables	Y
Parameters	a b s
Equations	ysq Y

The 2 Equations to Estimate
Y =	F(a(1), b(x), s)
ysq =	F(a, b, s)
Instruments	1 x

Nonlinear GMM Parameter Estimates
Parameter	Estimate	Approx Std Err	t Value	Approx Pr > \|t\|
a	2.065983	0.0657	31.45	<.0001
b	1.511075	0.0565	26.73	<.0001
s	1.483358	0.0498	29.78	<.0001

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