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); Johansen (1995a, 1995b). Only one COINTEG statement is allowed.

You specify the ECM= option in the MODEL statement or the COINTEG statement to fit the VECM( $p$ ). The P= option in the MODEL statement is used to specify the autoregressive order of the VECM.

The following statements are equivalent for fitting a VECM(2).

   proc varmax data=one;
      model y1-y3 / p=2 ecm=(rank=1);
   run;

   proc varmax data=one;
      model y1-y3 / p=2;
      cointeg rank=1;
   run;

To test restrictions of either $\balpha$ or $\bbeta$ or both, you specify either J= or H= or both, respectively. You specify the EXOGENEITY option in the COINTEG statement for tests of the weak exogeneity in the long-run parameters.

The following is an example of the COINTEG statement.

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

The following options can be used in the COINTEG statement:

EXOGENEITY

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

H=(matrix)

specifies the restrictions $H$ on the $k\times r$ or $(k+1)\times r$ cointegrated coefficient matrix $\bbeta$ such that $\bbeta = H\phi$ , where $H$ is known and $\phi$ is unknown. If the VECM( $p$ ) is specified with the COINTEG statement or with the ECM= option in the MODEL statement and the ECTREND option is not included with the ECM= specification, then the $H$ matrix has dimension $k\times m$ . If the VECM( $p$ ) is specified with the COINTEG statement or with the ECM= option in the MODEL statement and the ECTREND option is also used, then the $H$ matrix has dimension $(k+1)\times m$ . Here $k$ is the number of dependent variables, and $m$ is $r\leq m <k$ where $r$ is defined with the RANK= $r$ option.

For example, consider a system that contains four variables and the RANK=1 option with $\bbeta = (\beta _1, \beta _2, \beta _3, \beta _4)’$ . 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);

Here the matrix H is $4\times 3$ where $k=4$ and $m=3$ , and each row of the matrix H is separated by commas.

When the series has no separate deterministic trend, the constant term should be restricted by $\balpha ’_{\bot }\bdelta = 0$ . In the preceding example, the $\bbeta$ can be either $\bbeta = (\beta _1, \beta _2, \beta _3, \beta _4, 1)’$ or $\bbeta = (\beta _1, \beta _2, \beta _3, \beta _4, t)’$ . You can specify the restriction matrix for the previous test of $\beta _1 + \beta _2 = 0$ 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 dependent variables and the RANK=2 option, you can specify the restriction matrix for the test of $\beta _{1j}=-\beta _{2j} \; \mbox{for}\; j=1, 2$ as follows:

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

J=(matrix)

specifies the restrictions $J$ on the $k\times r$ adjustment matrix $\balpha$ such that $\balpha =J\psi$ , where $J$ is known and $\psi$ is unknown. The $k\times m$ matrix $J$ is specified by using this option, where $k$ is the number of dependent variables, $m$ is $r\leq m <k$ , and $r$ is defined with the RANK= $r$ option.

For example, when the system contains four variables and the RANK=1 option is used, you can specify the restriction matrix for the test of $\alpha _ j=0\; \mbox{for}\; j=2, 3, 4$ as follows:

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

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

   cointeg rank=2 j=(1 0, 0 0, 0 1);

NORMALIZE=variable

specifies a single dependent (endogenous) 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 dependent (endogenous) 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.