VARMACOV Call
computes the theoretical cross-covariance matrices for
a stationary VARMA() model
- CALL VARMACOV( cov, phi, theta, sigma <, p, q, lag>
);
The inputs to the VARMACOV subroutine are as follows:
- phi
- specifies a matrix, ,
containing the autoregressive coefficient matrices,
where is the number of elements in the subset
of the AR order and is the number of variables.
All the roots of should be greater than one in absolute value,
where is the finite order matrix polynomial in the backshift
operator , such that .
You must specify either phi or theta.
- theta
- specifies a matrix
containing 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.
- sigma
- specifies a 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 is defined as
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.
For example, consider a 4-dimensional vector time series,
and phi is a matrix.
If you specify p=1 (the default, since ),
the VARMACOV subroutine computes the theoretical
cross-covariance matrices of VAR(1) as
If you specify p=2, the VARMACOV subroutine computes
the cross-covariance matrices of VAR(2) as
Let phi be an matrix.
If you specify p, the VARMACOV subroutine computes
the cross-covariance matrices of VAR(3) as
If you do not specify p, the VARMACOV subroutine computes
the cross-covariance matrices of VAR(2) as
- q
- specifies the subset of the MA order. The quantity is defined as
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.
The usage of q is the same as that of p.
- lag
- specifies the length of lags, which must be a positive number.
If , the VARMACOV computes the cross-covariance matrices
from lag zero to lag .
By default, lag = 12.
The VARMACOV subroutine returns the following value:
- cov
- is a matrix that contains the theoretical
cross-covariance matrices of the VARMA() model.
Consider the following bivariate (
) VARMA(1,1) model:
To compute the cross-covariance matrices of this model, you can use the
following statements:
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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