VTSROOT Call :: SAS/IML(R) 12.1 User's Guide

VTSROOT Call

CALL VTSROOT (root, phi, theta <, p> <, q> ) ;

The VTSROOT subroutine computes the characteristic roots of the model from AR and MA characteristic functions.

The input arguments to the VTSROOT subroutine are as follows:

phi: specifies a $km_ p \times k$ matrix that contains 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 $km_ q \times k$ matrix that contains 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 $k(p_{max}+q_{max})\times 5$ 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

$\Phi =\left[\begin{matrix} 1.2 & -0.5 \cr 0.6 & 0.3 \cr \end{matrix}\right] ~ ~ \Theta =\left[\begin{matrix} -0.6 & 0.3 \cr 0.3 & 0.6 \cr \end{matrix}\right]$

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);
cols = {"Real" "Imag" "Modulus" "Radians" "Degrees"};
print root[colname=cols];

Figure 23.374: Characteristic Roots

root
Real	Imag	Modulus	Radians	Degrees
0.75	0.3122499	0.8124038	0.3945069	22.603583
0.75	-0.31225	0.8124038	-0.394507	-22.60358
0.6708204	0	0.6708204	0	0
-0.67082	0	0.6708204	3.1415927	180