The VARX(p,s) model can be written in the error correction form:
Let .
If and have full-rank r, and , then is an process.
If the condition fails and has reduced-rank where and are matrices with , then and are defined as matrices of full rank such that and .
If and have full-rank s, then the process is , which has the implication of model for the moving-average 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 (1995b).
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 full-rank r, let represent the model where and have full-rank s, and let represent the model where and have rank . The following table shows the relation between the models and the models.
Johansen (1995b) proposed the two-step 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 simulate an I(2) process and compute the rank test to test for cointegrated order 2:
proc iml; alpha = { 1, 1}; * alphaOrthogonal = { 1, -1}; beta = { 1, -0.5}; * betaOrthogonal = { 1, 2}; * alphaOrthogonal' * phiStar * betaOrthogonal = 0; phiStar = { 1 0, 0 0.5}; A1 = 2 * I(2) + alpha * beta` - phiStar; A2 = phiStar - I(2); phi = A1 // A2; sig = I(2); /* to simulate the vector time series */ call varmasim(y,phi) sigma=sig n=200 seed=2; cn = {'y1' 'y2'}; create simul4 from y[colname=cn]; append from y; close; quit; proc varmax data=simul4; model y1 y2 /noint p=2 cointtest=(johansen=(iorder=2)); run;
The last two columns in Figure 35.60 explain the cointegration rank test with integrated order 1. For a specified significance level, such as 5%, the output indicates that the null hypothesis that the series are not cointegrated (H0: ) is rejected, because the p-value for this test, shown in the column Pr > Trace of I(1), is less than 0.05. The results also indicate that the null hypothesis that there is a cointegrated relationship with cointegration rank 1 (H0: ) cannot be rejected at the 5% significance level, because the p-value for the test statistic, 0.7961, is greater than 0.05. Because of this latter result, the rows in the table that are associated with are further examined. The test statistic, 0.0257, tests the null hypothesis that the series are cointegrated order 2. The p-value that is associated with this test is 0.8955, which indicates that the null hypothesis cannot be rejected at the 5% significance level.