COINTEG Statement

The VARMAX Procedure

COINTEG Statement

COINTEG RANK= number < H= (matrix) > < J= (matrix) >
< EXOGENEITY > < NORMALIZE= variable > ;

The COINTEG statement fits the vector error correction model to the data, tests the restrictions of the long-run parameters and the adjustment parameters, and tests for the weak exogeneity in the long-run parameters. The cointegrated system uses the maximum likelihood analysis proposed by Johansen and Juselius (1990) and Johansen (1995a, 1995b). Only one COINTEG statement is allowed.

The VECM(p) form is written as

$\Delta y_{t}={\alpha} {\beta}' y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\delta} + {\epsilon}_t$

where ${\alpha}$ is a k×r matrix called adjustment coefficient, $r\leq k$ ; ${\beta}$ is a k×r matrix called the long-run coefficient; $\Delta$ means $\Delta y_{t}=y_{t}-y_{t-1}$ .

The following statements are examples of the COINTEG statement and the two examples are equivalent. You specify the ECM= option for fitting the VECM(p) with the P= option in the MODEL statement or the COINTEG statement. For testing of the restrictions of either ${\alpha}$ or ${\beta}$ or both, you specify either J= or H= or both. You specify the EXOGENEITY option for tests of the weak exogeneity in the long-run parameters.

            
   proc varmax data=one;
      model y1-y3 / p=2;
      cointeg rank=1 h=(1 0 0, -1 0 0, 0 1 0, 0 0 1)
              j=(1 0, 0 0, 0 1) exogeneity;

            
   proc varmax data=one;
      model y1-y3 / p=2 ecm=(rank=1);
      cointeg rank=1 h=(1 0 0, -1 0 0, 0 1 0, 0 0 1)
              j=(1 0, 0 0, 0 1) exogeneity;

EXOGENEITY

formulates the likelihood ratio tests for testing weak exogeneity in the long-run parameters. The hypothesis is that one variable is weakly exogenous for the others.

H= (matrix)

specifies the restrictions H on the k×r or (k+1)×r cointegrated coefficient matrix ${\beta}$ such that ${\beta}=H\phi$ , where H is known and $\phi$ is unknown. The k×m or (k+1)×m matrix H is specified using this option, where k is the number of endogenous variables, and m is $r\leq m \lt k$ with RANK=r. For example, consider that the system contains four variables and RANK=1 with ${\beta}=(\beta_1, \beta_2, \beta_3, \beta_4)'$ .The restriction matrix for the test of $\beta_1 + \beta_2=0$ can be specified as

            
   cointeg rank=1 h=(1 0 0, -1 0 0, 0 1 0, 0 0 1);

When the data have no deterministic trend, the constant term should be restricted by ${\alpha}'_{\bot}{\delta}=0$ . You can specify the restriction matrix for the previous test as follows:

   cointeg rank=1 
      h=(1 0 0 0, -1 0 0 0, 0 1 0 0, 0 0 1 0, 0 0 0 1);

When the cointegrated system contains three endogenous variables and RANK=2, you can specify the restriction matrix for the test of $\beta_{1j}=-\beta_{2j} \;{for}\;j=1,2$ as follows:

   cointeg rank=2 h=(1 0, -1 0, 0 1);

J= (matrix)

specifies the restrictions J on the k×r adjustment matrix ${\alpha}$ such that ${\alpha}=J\psi$ , where J is known and $\psi$ is unknown. The k×m matrix J is specified using this option, where k is the number of endogenous variables, and m is $r\leq m \lt k$ with RANK=r. For example, when the system contains four variables and RANK=1, you can specify the restriction matrix for the test of $\alpha_j=0\;{for}\;j=2,3,4$ as follows:

   cointeg rank=1 j=(1, 0, 0, 0);

When the system contains three variables and RANK=2, you can specify the restriction matrix for the test of $\alpha_{2j}=0\;{for}\;j=1,2$ as follows:

   cointeg rank=1 j=(1 0, 0 0, 0 1);

NORMALIZE= variable

specifies a single endogenous (dependent) variable name whose cointegrating vectors are normalized. If the variable name is different from that specified in the COINTTEST=(JOHANSEN= ) or ECM= option in the MODEL statement, the variable name specified in the COINTEG statement is used. If the normalized variable is not specified, cointegrating vectors are not normalized.

RANK= number

specifies the cointegration rank of the cointegrated system. This option is required in the COINTEG statement. The rank of cointegration should be greater than zero and less than the number of endogenous (dependent) variables. If the value of the RANK= option in the COINTEG statement is different from that specified in the ECM= option, the rank specified in the COINTEG statement is used.

Chapter Contents
Previous
Next
Top