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The VARMAX Procedure

Parameter Estimation and Testing on Restrictions

In the previous example, the VARX(1,0) model is written as

with

In Figure 30.20 of the preceding section, you can see several insignificant parameters. For example, the coefficients XL0_1_2, AR1_1_2, and AR1_3_2 are insignificant.

The following statements restrict the coefficients of for the VARX(1,0) model.

   /*--- Models with Restrictions and Tests ---*/
   
   proc varmax data=grunfeld;
      model y1-y3 = x1 x2 / p=1 print=(estimates);
      restrict XL(0,1,2)=0, AR(1,1,2)=0, AR(1,3,2)=0;
   run;

The output in Figure 30.21 shows that three parameters , , and are replaced by the restricted values, zeros. In the schematic representation of parameter estimates, the three restricted parameters , , and are replaced by .

Figure 30.21 Parameter Estimation with Restrictions

The VARMAX Procedure

XLag
Lag	Variable	x1	x2
0	y1	1.67592	0.00000
	y2	-6.30880	2.65308
	y3	-0.03576	-0.00919

AR
Lag	Variable	y1	y2	y3
1	y1	0.27671	0.00000	0.01747
	y2	-2.16968	0.10945	-0.93053
	y3	0.96398	0.00000	0.93412

Schematic Representation
Variable/Lag	C	XL0	AR1
y1	.	+*	.*.
y2	+	.+	..-
y3	-	..	+*+
+ is > 2*std error,  - is < -2*std error,  . is between,  * is N/A

The output in Figure 30.22 shows the estimates of the Lagrangian parameters and their significance. Based on the -values associated with the Lagrangian parameters, you cannot reject the null hypotheses , , and with the 0.05 significance level.

The TEST statement in the following example tests and for the VARX(1,0) model:

   proc varmax data=grunfeld;
      model y1-y3 = x1 x2 / p=1;
      test AR(1,3,1)=0;
      test XL(0,1,2)=0, AR(1,1,2)=0, AR(1,3,2)=0;
   run;

The output in Figure 30.23 shows that the first column in the output is the index corresponding to each TEST statement. You can reject the hypothesis test at the 0.05 significance level, but you cannot reject the joint hypothesis test at the 0.05 significance level.

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