## 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:

**
Output 18.16.1
PROC MODEL Output**

Y |

a b s(1) alpha(0.5) |

ysq ylagy Y |

F(a(1), b(x), s, alpha) |

F(a, b, s, alpha) |

F(a, b, s, alpha) |

1 x |

1.632798 |
0.1038 |
15.73 |
<.0001 |

1.513197 |
0.0698 |
21.67 |
<.0001 |

1.427888 |
0.0984 |
14.52 |
<.0001 |

0.543985 |
0.0809 |
6.72 |
<.0001 |