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 $\mr {E} (\bepsilon _ t \bepsilon _ t’)=\Sigma$ is assumed to be positive-definite, there exists a lower triangular matrix $A_0$ with ones on the diagonals such that $A_0\Sigma A_0’=\Sigma ^ d$ , where $\Sigma ^ d$ is a diagonal matrix with positive diagonal elements.

$\begin{eqnarray*} A_0 \mb {y} _ t = \sum _{i=1}^{p} A_ i\mb {y} _{t-i} + \sum _{i=0}^{s}C_ i^*\mb {x} _{t-i} + C_0\bepsilon _ t - \sum _{i=1}^{q}C_ i\bepsilon _{t-i} \end{eqnarray*}$

where $A_ i = A_0 \Phi _ i$ , $C_ i^* = A_0 \Theta _ i^*$ , $C_0 = A_0$ , and $C_ i = A_0 \Theta _ i$ .

As an alternative form,

$\begin{eqnarray*} A_0 \mb {y} _ t = \sum _{i=1}^{p} A_ i\mb {y} _{t-i} + \sum _{i=0}^{s}C_ i^*\mb {x} _{t-i} + \mb {a}_ t - \sum _{i=1}^{q}C_ i\mb {a}_{t-i} \end{eqnarray*}$

where $A_ i = A_0 \Phi _ i$ , $C_ i^* = A_0 \Theta _ i^*$ , $C_ i = A_0 \Theta _ i A_0^{-1}$ , and $\mb {a}_ t = C_0\bepsilon _ t$ has a diagonal covariance matrix $\Sigma ^ d$ . 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 $\mb {y} _ t$ at lag 0 or a leading coefficient matrix for $\bepsilon _ t$ 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 $A_0 \mb {y} _ t$ , 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 $\bepsilon _ t$ estimated from the VARMAX model form is

$\begin{eqnarray*} \Sigma _{\bepsilon } = \left( \begin{array}{rr} 1.28875 & 0.39751 \\ 0.39751 & 1.41839 \end{array} \right) \end{eqnarray*}$

Figure 35.25 shows the results from estimating the model as a dynamic simultaneous equations model. By the decomposition of $\Sigma _{\bepsilon }$ , you get a diagonal matrix ( $\Sigma _\mb {{a}}$ ) and a lower triangular matrix ( $A_0$ ) such as $\Sigma _\mb {{a}} = A_0 \Sigma _{\bepsilon } A_0’$ where

$\begin{eqnarray*} \Sigma _\mb {{a}} = \left( \begin{array}{rr} 1.28875 & 0 \\ 0 & 1.29578 \end{array} \right) ~ ~ \mr {and} ~ ~ A_0 =\left( \begin{array}{rr} 1 & 0 \\ -0.30845 & 1 \end{array} \right) \end{eqnarray*}$

The lower triangular matrix ( $A_0$ ) 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

$\begin{eqnarray*} \left( \begin{array}{rr} 1 & 0 \\ -0.30845 & 1 \end{array} \right) \mb {y} _ t = \left( \begin{array}{rr} 1.15977 & -0.51058 \\ 0.18861 & 0.54247 \end{array} \right) \mb {y} _{t-1} + \mb {a}_ t \end{eqnarray*}$

The resulting two-equation system can be written as

$\begin{eqnarray*} y_{1t} & =& 1.15977 y_{1,t-1} - 0.51058 y_{2,t-1} + a_{1t} \\ y_{2t} & =& 0.30845 y_{1t} + 0.18861 y_{1,t-1} + 0.54247 y_{2,t-1} + a_{2t} \\ \end{eqnarray*}$