VARMACOV Call

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VARMACOV Call

computes the theoritical auto-cross covariance matrices for stationary VARMA(p,q) model

CALL VARMACOV( cov, phi, theta, sigma <, p, q, lag>);

The inputs to the VARMACOV subroutine are as follows:

phi

specifies to a kp ×k matrix containing the vector autoregressive coefficient matrices. All the roots of $|\Phi(B)|=0$ are greater than one in absolute value.

theta

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

sigma

specifies a k ×k symmetric positive-definite covariance matrix of the innovation series. By default, sigma is an identity matrix with dimension k.

p

specifies the order of AR. You can also specify the subset of the order of AR. By default, let $phi = \Phi$ ,

$p={ {\rm the number of row of matrix} \Phi \over {\rm the number of column of matrix} \Phi }.$

For example, consider a 4 dimensional vector time series, if $phi = \Phi$ is 4 ×4 matrix and p=1, the VARMACOV subroutine computes the theoritical auto-cross covariance matrices of VAR(1) as follows

$y_t = \Phi y_{t-1} + m{\epsilon}_t.$

If $phi = \Phi$ is 4×4 matrix and p=2, the VARMACOV subroutine computes the theoritical auto-cross covariance matrices of VAR(2) as follows

$y_t = \Phi y_{t-2} + m{\epsilon}_t.$

If $phi = [ \Phi_1' \Phi_3' ]'$ is 8×4 matrix and p = {1,3 }, the VARMACOV subroutine computes the theoritical auto-cross covariance matrices of VAR(3) as follows

$y_t = \Phi_1 y_{t-1} + \Phi_3 y_{t-3} + m{\epsilon}_t.$

q

specifies the order of MA. You can specify the subset of the order of MA. By default, let $theta = \Theta$ ,

$q={ {\rm the number of row of matrix} \Theta \over {\rm the number of column of matrix} \Theta }.$

The usage of q is the same as that of p.

lag

specifies the length of lags, which must be a positive number. If lag = h, the VARMACOV computes the auto-cross covariance matrices from at lag zero to at lag h. By default, lag = 12.

The VARMACOV subroutine returns the following value:

cov: refers an (k*lag)×k matrices the theoritical auto-cross covariance VARMA(p,q) series. In case of VMA(q) when p=0, the VARMACOV computes the auto-cross covariance matrices from at lag zero to at lag q.

To compute the theoritical auto-cross covariance matrices of a bivariate (k=2) VARMA(1,1) model

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

with $m{\epsilon}_t \sim WN( 0, \Sigma)$ ,where

$\Sigma=[\matrix{1.0 & 0.5 \cr 0.5 & 1.25\cr }], \Phi=[\matrix{1.2 & -0.5 \cr 0.6 & 0.3 \cr }], \Theta=[\matrix{-0.6 & 0.3 \cr 0.3 & 0.6 \cr }],$

you can specify

  call varmacov(cov, phi, theta, sigma) lag=5;

The VARMACOV subroutine computes theoritical auto-cross covariance matrices for the VARMA(p,q) model when AR coefficient matrices $\Phi_i$ , MA coefficient matrices $\Theta_i$ , and an inovation covariance matrix $\Sigma$ are known. Auto-cross covariance matrices $\Gamma(l)$ are

$\Gamma(l) = \sum_{j=1}^p \Gamma(l-j) \Phi_j' - \sum_{j=l}^q \Psi(j-l) \Sigma \Theta_j', {\rm for}l=0, ... ,q$

$\Gamma(l) = \sum_{j=1}^p \Gamma(l-j) \Phi_j' {\rm for} l\gt q$

where $\Psi_j$ satisfy

$\Psi_j = \Phi_1 \Psi_{j-1}+\Phi_2 \Psi_{j-2}+ ... +\Phi_p \Psi_{j-p}-\Theta_j$

with $\Theta_0 = -I$ , $\Psi_0 = I$ , and $\Psi_j = 0$ for j < 0.

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