Language Reference: VTSROOT Call :: SAS/IML(R) 9.2 User's Guide

Language Reference

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 a matrix containing the autoregressive coefficient matrices, where is the number of the elements in the subset of the AR order and $k\geq 2$ is the number of variables. You must specify either phi or theta.
theta: specifies a matrix containing the moving-average coefficient matrices, where is the number of the elements in the subset of the MA order. You must specify either phi or theta.
p: specifies the subset of the AR order. See the VARMACOV subroutine.
q: specifies the subset of the MA order. See the VARMACOV subroutine.

The VTSROOT subroutine returns the following value:

root: is a $k(p_{max}+q_{max})x 5$ matrix, where $p_{max}$ is the maximum order of the AR characteristic function and $q_{max}$ is the maximum order of the MA characteristic function. The first $k p_{max}$ rows refer to the results of the AR characteristic function; the last $k q_{max}$ rows refer to the results of the MA characteristic function.

The first column contains the real parts, , of eigenvalues of companion matrix associated with the AR( $p_{max}$ ) or MA( $q_{max}$ ) characteristic function; the second column contains the imaginary parts, , of the eigenvalues; the third column contains the moduli of the eigenvalues, $\sqrt{x^2+y^2}$ ; the fourth column contains the arguments ( $\arctan(y/x)$ ) of the eigenvalues, measured in radians from the positive real axis. The fifth column contains the arguments expressed in degrees rather than radians.

Consider the roots of the characteristic functions, $\phi(b)=i-\phi b$ and $\theta(b)=i-\theta b$ , where

is an identity matrix with dimension 2 and

$\phi=[\matrix{1.2 & -0.5 \cr 0.6 & 0.3 \cr }] \theta=[\matrix{-0.6 & 0.3 \cr 0.3 & 0.6 \cr }]$

To compute these roots, you can use the following statements:

  
   phi  = { 1.2 -0.5, 0.6 0.3 }; 
   theta= {-0.6  0.3, 0.3 0.6 }; 
   call vtsroot(root, phi, theta);