The VARX(,) model can be written in the error correction form:

Let .
If and have fullrank , and , then is an process.
If the condition fails and has reducedrank where and are matrices with , then and are defined as matrices of full rank such that and .
If and have fullrank , then the process is , which has the implication of model for the movingaverage representation.

The matrices , , and are determined by the cointegration properties of the process, and and are determined by the initial values. For details, see Johansen (1995a).
The implication of the model for the autoregressive representation is given by

where and .
The cointegrated model is given by the following parameter restrictions:

where and are matrices with . Let represent the model where and have fullrank , let represent the model where and have fullrank , and let represent the model where and have rank . The following table shows the relation between the models and the models.
Table 36.2: Relation between the and Models




k 
k1 



0 











1 























Johansen (1995a) proposed the twostep procedure to analyze the model. In the first step, the values of are estimated using the reduced rank regression analysis, performing the regression analysis , , and on and . This gives residuals , , and , and residual product moment matrices

Perform the reduced rank regression analysis on corrected for , and , and solve the eigenvalue problem of the equation

where for .
In the second step, if are known, the values of are determined using the reduced rank regression analysis, regressing on corrected for , and .
The reduced rank regression analysis reduces to the solution of an eigenvalue problem for the equation

where









where .
The solution gives eigenvalues and eigenvectors . Then, the ML estimators are






The likelihood ratio test for the reduced rank model with rank in the model is given by

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 36.60 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 . 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 36.60: Cointegrated I(2) Test (IORDER= Option)
Cointegration Rank Test for I(2)  

r\krs  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 