The VARMAX Procedure

Dynamic Simultaneous Equations Modeling

In the econometrics literature, the VARMAX(p,q,s) 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(p,q,s) 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.

Figure 35.25: Dynamic Simultaneous Equations (DYNAMIC Option)

The VARMAX Procedure

Covariances of Innovations
Variable y1 y2
y1 1.28875 0.00000
y2 0.00000 1.29578

AR
Lag Variable y1 y2
0 y1 1.00000 0.00000
y2 -0.30845 1.00000
1 y1 1.15977 -0.51058
y2 0.18861 0.54247

Dynamic Model Parameter Estimates
Equation Parameter Estimate Standard
Error
t Value Pr > |t| Variable
y1 AR1_1_1 1.15977 0.05508 21.06 0.0001 y1(t-1)
AR1_1_2 -0.51058 0.07140 -7.15 0.0001 y2(t-1)
y2 AR0_2_1 0.30845       y1(t)
AR1_2_1 0.18861 0.05779 3.26 0.0015 y1(t-1)
AR1_2_2 0.54247 0.07491 7.24 0.0001 y2(t-1)

In Figure 35.4 in the section Getting Started: VARMAX Procedure, the covariance of estimated from the VARMAX model form is Figure 35.25 shows the results from estimating the model as a dynamic simultaneous equations model. By the decomposition of , you get a diagonal matrix ( ) and a lower triangular matrix ( ) such as where The lower triangular matrix ( ) is shown in the left side of the simultaneous equations model. The parameter estimates in equations system are shown in the right side of the two-equations system.

The simultaneous equations model is written as The resulting two-equation system can be written as 