## VTSROOT Call

**calculates the characteristic roots of the model
from AR and MA characteristic functions**
**CALL VTSROOT(** *root, phi, theta<, p, q>***);**

The inputs to the VTSROOT subroutine are as follows:
*phi*
- specifies to a
*kp* ×*k* matrix
containing the vector autoregressive coefficient matrices.

*theta*
- specifies to a
*kq* ×*k* matrix
containing the vector moving-average coefficient matrices.
You must specify either *phi* or *theta*.

*p*
- specifies the order of VAR. See the VARMACOV subroutine.

*q*
- specifies the order of VMA. See the VARMACOV subroutine.

The VTSROOT subroutine returns the following value:
*root*
- refers an
*k**(*maxar*+*maxma*)×5 matrices, where *maxar* and *maxma*
are the maximum orders corresponding to AR and MA, respectively.
The first *k***maxar* rows refer the results of the AR part, and
the last *k***maxma* rows refer the results of the MA part.
The first column refers the real part of eigenvalues of
companion matrices associated with the VAR(*p*) characteristic function.
The second column refers the imaginary part of eigenvalues.
The third column refers the modulus of eigenvalues.
The forth column refers the degree(radian) of eigenvalues.
The fifth column refers the degree(radian×180) of eigenvalues.

To compute the roots of a bivariate(*k*=2) VARMA(1,1) model

where

you can specify
call vtsroot(root, phi, theta);

The VTSROOT subroutine computes the eigenvalues of the *kp* ×*kp*
companion matrices associated with the AR(p) characteristic function, where
*k* is the number of dependent variables.
They indicate the stationary condition of
the process since the stationary condition
on the roots of in the AR(*p*)
is equivalent to the condition in the corresponding AR(1) representation
that all eigenvalues of the
companion matrix be less than one in absolute value.
The stationarity condition is equivalent to the condition in the
corresponding AR(1) representation,
,that all eigenvalues of the
*kp* ×*kp* companion matrix be less than one in absolute value,
where **Y**_{t} = (**y**_{t}', ... ,**y**_{t-p+1}')',
,and

Similarly, it can apply for checking the invertible condition of the MA
proces.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.