VARMACOV Call

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Changes and Enhancements

VARMACOV Call

computes the theoretical cross-covariance matrices for a 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 a km_p ×k matrix, $\Phi$ , containing the autoregressive coefficient matrices, where m_p is the number of elements in the subset of the AR order and $k\geq 2$ is the number of variables. All the roots of $|\Phi(B)|=0$ should be greater than one in absolute value, where $\Phi(B)$ is the finite order matrix polynomial in the backshift operator B, such that B^jy_t = y_t-j. You must specify either phi or theta.

theta

specifies a km_q ×k matrix containing 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.

sigma

specifies a k ×k symmetric positive-definite covariance matrix of the innovation series. If sigma is not specified, then an identity matrix is used.

p

specifies the subset of the AR order. The quantity m_p is defined as

m_p= nrow(phi)/ncol(phi)

where nrow(phi) is the number of rows of the matrix phi and ncol(phi) is the number of columns of the matrix phi.

If you do not specify p, the default subset is p= {1,2, ... ,m_p}.

For example, consider a four-dimensional vector time series, and phi is a 4 ×4 matrix. If you specify p=1 (the default, since m_p=4/4=1), the VARMACOV subroutine computes the theoretical cross-covariance matrices of VAR(1) as $y_t=\Phi y_{t-1} + {\epsilon}_t.$

If you specify p=2, the VARMACOV subroutine computes the cross-covariance matrices of VAR(2) as $y_t=\Phi y_{t-2} + {\epsilon}_t.$

Let phi $=[ \Phi_1', \Phi_2' ]'$ be an 8×4 matrix. If you specify p= {1,3}, the VARMACOV subroutine computes the cross-covariance matrices of VAR(3) as $y_t=\Phi_1{y}_{t-1}+\Phi_2{y}_{t-3}+{\epsilon}_t.$ If you do not specify p, the VARMACOV subroutine computes the cross-covariance matrices of VAR(2) as $y_t=\Phi_1{y}_{t-1}+\Phi_2 y_{t-2}+{\epsilon}_t.$

q

specifies the subset of the MA order. The quantity m_q is defined as

m_q= nrow(theta)/ncol(theta)

where nrow(theta) is the number of rows of matrix theta and ncol(theta) is the number of columns of matrix theta.

If you do not specify q, the default subset is q= {1,2, ... ,m_q}.

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 cross-covariance matrices from lag zero to lag h. By default, lag = 12.

The VARMACOV subroutine returns the following value:

cov: is a k(lag+1)×k matrix that contains the theoretical cross-covariance matrices of the VARMA(p,q) model.

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

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

where

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

you can specify

  phi  = { 1.2 -0.5, 0.6 0.3 };
  theta= {-0.6  0.3, 0.3 0.6 };
  sigma= { 1.0  0.5, 0.5 1.25};
  call varmacov(cov, phi, theta, sigma) lag=5;

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