The VARMAX Procedure

Parameter Estimation and Testing on Restrictions

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

$\displaystyle  \mb {y} _{t} = \bdelta + \Theta ^{*}_0\mb {x} _{t} + \Phi _1\mb {y} _{t-1} + \bepsilon _ t  $

with

$\displaystyle  \Theta _0^{*} = \left( \begin{array}{rr} \theta ^{*}_{11} &  \theta ^{*}_{12} \\ \theta ^{*}_{21} &  \theta ^{*}_{22} \\ \theta ^{*}_{31} &  \theta ^{*}_{32} \end{array} \right) ~ ~ ~  \Phi _1 = \left( \begin{array}{rrr} \phi _{11} &  \phi _{12} &  \phi _{13} \\ \phi _{21} &  \phi _{22} &  \phi _{23} \\ \phi _{31} &  \phi _{32} &  \phi _{33} \end{array} \right)  $

In Figure 36.20 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 $\theta _{12}^*=\phi _{12}=\phi _{32}=0$ 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 36.21 shows that three parameters $\theta _{12}^*$, $\phi _{12}$, and $\phi _{32}$ are replaced by the restricted values, zeros. In the schematic representation of parameter estimates, the three restricted parameters $\theta _{12}^*$, $\phi _{12}$, and $\phi _{32}$ are replaced by $*$.

Figure 36.21: 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 36.22 shows the estimates of the Lagrangian parameters and their significance. Based on the $p$-values associated with the Lagrangian parameters, you cannot reject the null hypotheses $\theta _{12}^*=0$, $\phi _{12}=0$, and $\phi _{32}=0$ with the 0.05 significance level.

Figure 36.22: RESTRICT Statement Results

Testing of the Restricted Parameters
Parameter Estimate Standard
Error
t Value Pr > |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


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 output in Figure 36.23 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, but you cannot reject the joint hypothesis test $\theta _{12}^*=\phi _{12}=\phi _{32}=0$ at the 0.05 significance level.

Figure 36.23: TEST Statement Results

The VARMAX Procedure

Testing of the Parameters
Test DF Chi-Square Pr > ChiSq
1 1 150.31 <.0001
2 3 0.34 0.9522