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 $m_ p$ 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 $m_ q$ 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, $x$, of eigenvalues of companion matrix associated with the AR($p_{max}$) or MA($q_{max}$) characteristic function; the second column contains the imaginary parts, $y$, 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 $I$ 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 24.429: 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