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 longrun parameters
and the adjustment parameters, and tests for the weak exogeneity in the longrun 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.
You specify the ECM= option in the MODEL statement or the COINTEG statement to fit the VECM(). 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 y1y3 / p=2 ecm=(rank=1);
run;
proc varmax data=one;
model y1y3 / p=2;
cointeg rank=1;
run;
To test restrictions of either or 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 longrun parameters.
The following is an example of the COINTEG statement.
proc varmax data=one;
model y1y3 / 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 longrun parameters. The null hypothesis is that
one variable is weakly exogenous for the others.

H=(matrix)

specifies the restrictions on the or cointegrated coefficient matrix such that , where is known and is unknown. If the VECM() 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 matrix has dimension . If the VECM() is specified with the COINTEG statement or with the ECM= option in the MODEL statement and the ECTREND option is also used,
then the matrix has dimension . Here is the number of dependent variables, and is where is defined with the RANK= option.
For example, consider a system that contains four variables and the RANK=1 option with . The restriction matrix for the test of 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 where and , 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 . In the preceding example, the can be either or . You can specify the restriction matrix for the previous test of 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 as follows:
cointeg rank=2 h=(1 0, 1 0, 0 1);

J=(matrix)

specifies the restrictions on the adjustment matrix such that , where is known and is unknown. The matrix is specified by using this option, where is the number of dependent variables, is , and is defined with the RANK= 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 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 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.