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.