The VARMAX Procedure

The BOUND statement sets up linear bounds for parameters when the maximum likelihood method is applied to the estimation of VARMAX, VECM, VARMAX-GARCH, and VEC-ARMAX-GARCH models. Only one BOUND statement is allowed. If you specify more than one restriction, separate them with commas. The restrictions are specified in the same manner as the restrictions in the RESTRICT statement. For information about how to define restrictions by using matrix expressions, operators, and functions, see the section RESTRICT Statement. Both equality and inequality constraints are allowed in the BOUND statement, although usually equality constraints are specified in the RESTRICT statement and inequality constraints are specified in the BOUND statement.

To use the BOUND statement, you need to know the form of the model. If you do not specify the GARCH statement, the COINTEG statement, or the ECM=, P=, Q=, or XLAG= option in the MODEL statement, then the BOUND statement is not applicable. If you specify the ECM=(NORMALIZE=), METHOD=LS, or PRIOR= option in the MODEL statement, or if you specify the EXOGENEITY, H=, J=, or NORMALIZE= option in the COINTEG statement, the BOUND statement is ignored. Nonlinear restrictions on parameters are not supported.

The following is an example of the BOUND statement for a bivariate (k=2) zero-mean VARMA(1,1) model, which is by default estimated by maximum likelihood method because the MA term is present:

proc varmax data=one;
   model y1 y2 / noint p=1 q=1;
   bound -1<=AR<=1, 0<MA;
run;

This BOUND statement specifies that all AR parameters must be between and 1 and that all MA parameters must be positive.

You can use the BOUND statement together with the RESTRICT statement, as in the following bivariate (k=2) zero-mean VARMA(1,1) model:

proc varmax data=one;
   model y1 y2 / noint p=1 q=1;
   bound AR+MA>=0.001;
   restrict AR(1,1,2) = 0.5;
run;