In the econometrics literature, the VARMAX(,,) model is sometimes written in a form that is slightly different than the one shown in the previous section. This alternative form is referred to as a dynamic simultaneous equations model or a dynamic structural equations model.

Since is assumed to be positive-definite, there exists a lower triangular matrix with ones on the diagonals such that , where is a diagonal matrix with positive diagonal elements.

where , , , and .

As an alternative form,

where , , , and has a diagonal covariance matrix . The PRINT=(DYNAMIC) option returns the parameter estimates that result from estimating the model in this form.

A dynamic simultaneous equations model involves a leading (lower triangular) coefficient matrix for at lag 0 or a leading coefficient matrix for at lag 0. Such a representation of the VARMAX(,,) model can be more useful in certain circumstances than the standard representation. From the linear combination of the dependent variables obtained by , you can easily see the relationship between the dependent variables in the current time.

The following statements provide the dynamic simultaneous equations of the VAR(1) model.

proc iml;
   sig = {1.0  0.5, 0.5 1.25};
   phi = {1.2 -0.5, 0.6 0.3};
   /* simulate the vector time series */
   call varmasim(y,phi) sigma = sig n = 100 seed = 34657;
   cn = {'y1' 'y2'};
   create simul1 from y[colname=cn];
   append from y;
quit;

data simul1;
   set simul1;
   date = intnx( 'year', '01jan1900'd, _n_-1 );
   format date year4.;
run;

proc varmax data=simul1;
   model y1 y2 / p=1 noint print=(dynamic);
run;

This is the same data set and model used in the section Getting Started: VARMAX Procedure. You can compare the results of the VARMA model form and the dynamic simultaneous equations model form.