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
,
![p={ {\rm the number of row of matrix} \Phi \over
{\rm the number of column of matrix} \Phi }.](images/i17eq396.gif)
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
![y_t = \Phi y_{t-1} + m{\epsilon}_t.](images/i17eq397.gif)
If
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.](images/i17eq398.gif)
If
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.](images/i17eq400.gif)
- q
- specifies the order of MA. You can specify the subset of the order of MA.
By default, let
,
![q={ {\rm the number of row of matrix} \Theta \over
{\rm the number of column of matrix} \Theta }.](images/i17eq402.gif)
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},](images/i17eq403.gif)
with
,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
}],](images/i17eq405.gif)
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
![\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](images/i17eq410.gif)
![\Gamma(l) = \sum_{j=1}^p \Gamma(l-j) \Phi_j' {\rm for} l\gt q](images/i17eq411.gif)
where
satisfy
![\Psi_j = \Phi_1 \Psi_{j-1}+\Phi_2 \Psi_{j-2}+ ...
+\Phi_p \Psi_{j-p}-\Theta_j](images/i17eq413.gif)
with
,
, and
for j < 0.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.