The VARMAX Procedure

I(2) Model

The VARX( $\text{[math]}$ , $\text{[math]}$ ) model can be written in the error correction form:

$\text{[math]}$

Let $\text{[math]}$ .

If $\text{[math]}$ and $\text{[math]}$ have full-rank $\text{[math]}$ , and $\text{[math]}$ , then $\text{[math]}$ is an $\text{[math]}$ process.

If the condition $\text{[math]}$ fails and $\text{[math]}$ has reduced-rank $\text{[math]}$ where $\text{[math]}$ and $\text{[math]}$ are $\text{[math]}$ matrices with $\text{[math]}$ , then $\text{[math]}$ and $\text{[math]}$ are defined as $\text{[math]}$ matrices of full rank such that $\text{[math]}$ and $\text{[math]}$ .

If $\text{[math]}$ and $\text{[math]}$ have full-rank $\text{[math]}$ , then the process $\text{[math]}$ is $\text{[math]}$ , which has the implication of $\text{[math]}$ model for the moving-average representation.

$\text{[math]}$

The matrices $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ are determined by the cointegration properties of the process, and $\text{[math]}$ and $\text{[math]}$ are determined by the initial values. For details, see Johansen (1995a).

The implication of the $\text{[math]}$ model for the autoregressive representation is given by

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ .

Test for I(2)

The $\text{[math]}$ cointegrated model is given by the following parameter restrictions:

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ are $\text{[math]}$ matrices with $\text{[math]}$ . Let $\text{[math]}$ represent the $\text{[math]}$ model where $\text{[math]}$ and $\text{[math]}$ have full-rank $\text{[math]}$ , let $\text{[math]}$ represent the $\text{[math]}$ model where $\text{[math]}$ and $\text{[math]}$ have full-rank $\text{[math]}$ , and let $\text{[math]}$ represent the $\text{[math]}$ model where $\text{[math]}$ and $\text{[math]}$ have rank $\text{[math]}$ . The following table shows the relation between the $\text{[math]}$ models and the $\text{[math]}$ models.

Table 35.2 Relation between the $\text{[math]}$ and $\text{[math]}$ Models
					$\text{[math]}$					$\text{[math]}$
$\text{[math]}$	k		k-1		$\text{[math]}$		$\text{[math]}$
0	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
1			$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$							$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$							$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

Johansen (1995a) proposed the two-step procedure to analyze the $\text{[math]}$ model. In the first step, the values of $\text{[math]}$ are estimated using the reduced rank regression analysis, performing the regression analysis $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ on $\text{[math]}$ and $\text{[math]}$ . This gives residuals $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ , and residual product moment matrices

$\text{[math]}$

Perform the reduced rank regression analysis $\text{[math]}$ on $\text{[math]}$ corrected for $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ , and solve the eigenvalue problem of the equation

$\text{[math]}$

where $\text{[math]}$ for $\text{[math]}$ .

In the second step, if $\text{[math]}$ are known, the values of $\text{[math]}$ are determined using the reduced rank regression analysis, regressing $\text{[math]}$ on $\text{[math]}$ corrected for $\text{[math]}$ , and $\text{[math]}$ .

The reduced rank regression analysis reduces to the solution of an eigenvalue problem for the equation

$\text{[math]}$

where

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

where $\text{[math]}$ .

The solution gives eigenvalues $\text{[math]}$ and eigenvectors $\text{[math]}$ . Then, the ML estimators are

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The likelihood ratio test for the reduced rank model $\text{[math]}$ with rank $\text{[math]}$ in the model $\text{[math]}$ is given by

$\text{[math]}$

The following statements compute the rank test to test for cointegrated order 2:

proc varmax data=simul2;
   model y1 y2 / p=2 cointtest=(johansen=(iorder=2));
run;

The last two columns in Figure 35.60 explain the cointegration rank test with integrated order 1. The results indicate that there is the cointegrated relationship with the cointegration rank 1 with respect to the 0.05 significance level because the test statistic of 0.5552 is smaller than the critical value of 3.84. Now, look at the row associated with $\text{[math]}$ . Compare the test statistic value, 211.84512, to the critical value, 3.84, for the cointegrated order 2. There is no evidence that the series are integrated order 2 at the 0.05 significance level.

Figure 35.60 Cointegrated I(2) Test (IORDER= Option)

The VARMAX Procedure

Cointegration Rank Test for I(2)
r\k-r-s	2	1	Trace of I(1)	5% CV of I(1)
0	720.40735	308.69199	61.7522	15.34
1		211.84512	0.5552	3.84
5% CV I(2)	15.34000	3.84000