Example 19.15 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:

$y = a + b x + \epsilon , \, \epsilon \sim iid \, N(0, s^2)$

In the following SAS statements, $ysim$ is simulated, and the first moment and the second moment of $ysim$ are compared with those of the observed endogenous variable $y$ .

title "Simple regression model";

data regdata;
   do i=1 to 500;
      x = rannor( 1013 );
      Y = 2 + 1.5 * x + 1.5 * rannor( 1013 );
      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;

Alternatively, the MOMENT statement can be used to specify the moments using the following syntax:

proc model data=regdata;
   parms a b s;
   instrument x;

   ysim = (a+b*x) + s * rannor( 8003 );
   y = ysim;
   moment y = (2);

   fit y / gmm ndraw;
   bound s > 0;
run;

The output of the MODEL procedure is shown in Output 19.15.1:

Output 19.15.1: 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