Parameter Estimation and Testing on Restrictions

Chapter Contents

The VARMAX Procedure

Parameter Estimation and Testing on Restrictions

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

$y_{t}={\delta} + \Theta^{*}_0{x}_{t} + \Phi_1{y}_{t-1} + {\epsilon}_t$

with

$\Theta_0^{*}=( \theta^{*}_{11} & \theta^{*}_{12} \ \theta^{*}_{21} & \theta^... ... \phi_{21} & \phi_{22} & \phi_{23} \ \phi_{31} & \phi_{32} & \phi_{33} ).$

In Figure 4.19, you can see the coefficients XL_0_1_2, AR_1_1_2, and AR_1_3_2 are insignificant. The following statements restrict the coefficients of $\theta_{12}^*=\phi_{12}=\phi_{32}=0$ for the VARX(1,0) model.

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

The VARMAX Procedure

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

AR Coefficient Estimates
Lag	Variable	y1	y2	y3
1	y1	0.27671	0	0.01747
	y2	-2.16968	0.10945	-0.93053
	y3	0.96398	0	0.93412

Figure 4.20: Parameter Estimation on Restrictions

The output in Figure 4.20 shows that three parameters $\theta_{12}^*$ , $\phi_{12}$ , and $\phi_{32}$ are replaced by the restricted values.

The VARMAX Procedure

Restriction Results
Parameter	Lagrange Multiplier	Std Error	T Ratio	Prob>\|T\|
XL0_1_2	1.74969	21.44026	0.08	0.9389
AR1_1_2	30.36254	70.74347	0.43	0.6899
AR1_3_2	55.42191	164.03075	0.34	0.7524

Figure 4.21: RESTRICT Statement Results

The output in Figure 4.21 shows the estimates of the Lagrangian parameters and their significances. You cannot reject the null hypotheses $\theta_{12}^*=0$ , $\phi_{12}=0$ , and $\phi_{32}=0$ with the 0.05 significance level.

The TEST statement in the following example tests $\phi_{31}=0$ and $\theta_{12}^*=\phi_{12}=\phi_{32}=0$ 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 VARMAX Procedure

Test Results
Test	Chi-Square	DF	Prob>Chisq
1	150.31	1	<.0001
2	0.34	3	0.9522

Figure 4.22: TEST Statement Results

The output in Figure 4.22 shows that the first column in the output is the index corresponding to each TEST statement; you can reject the hypothesis test $\phi_{31}=0$ at the 0.05 significance level; you cannot reject the joint hypothesis test $\theta_{12}^*=\phi_{12}=\phi_{32}=0$ at the 0.05 significance level.

Chapter Contents
Previous
Next
Top