## 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 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 ,

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

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

If is 8×4 matrix
and *p* = {1,3 }, the VARMACOV subroutine computes the theoritical auto-cross covariance
matrices of VAR(3) as follows

*q*
- specifies the order of MA. You can specify the subset of the order of MA.
By default, let ,

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

with ,where

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 , MA coefficient matrices
, and an inovation covariance matrix are known.
Auto-cross covariance matrices are

where satisfy

with , , and for *j* < 0.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.