VTSROOT Call

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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

$y_t = \Phi y_{t-1} + m{\epsilon}_t - \Theta m{\epsilon}_{t-1},$

where

$\Phi=[\matrix{1.2 & -0.5 \cr 0.6 & 0.3 \cr }], \Theta=[\matrix{-0.6 & 0.3 \cr 0.3 & 0.6 \cr }],$

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 $|\Phi(B)|=0$ 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, $Y_t = m{\Phi}Y_{t-1} + m{\varepsilon}_t$ ,that all eigenvalues of the kp ×kp companion matrix $\Phi$ be less than one in absolute value, where Y_t = (y_t', ... ,y_t-p+1')', $m{\varepsilon}_t = ( m{\epsilon}_t', 0', ... , 0')'$ ,and

$m{\Phi} = [ \matrix{ \Phi_1 & \Phi_2 & ... & \Phi_{p-1} & \Phi_{p} \cr I & 0... ... \vdots & \vdots & \ddots & \vdots & \vdots \cr 0 & 0 & ... & I & 0 \cr } ]$

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

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