g05hdf
g05hdf
© Numerical Algorithms Group, 2002.
Purpose
G05HDF Generates a realisation of a multivariate time series from
a VARMA model
Synopsis
[w,ref,ifail] = g05hdf(ip,iq,par,qq,n<,mean,ref,mode,ifail>)
Description
T
Let the vector W =(w ,w ,...,w ) , denote a k dimensional time
t 1t 2t kt
series which is assumed to follow a vector autoregressive moving
average (VARMA) model of the form:
W -(mu)=(phi) (W -(mu))+(phi) (W -(mu))+...+(phi) (W -(mu))+
t 1 t-1 2 t-2 p t-p
(epsilon) -(theta) (epsilon) -(theta) (epsilon) -...
t 1 t-1 2 t-2
-(theta) (epsilon) (1)
q t-q
T
where (epsilon) =((epsilon) ,(epsilon) ,...,(epsilon) ) , is a
t 1t 2t kt
vector of k residual series assumed to be Normally distributed
with zero mean and positive-definite covariance matrix (Sigma).
The components of (epsilon) are assumed to be uncorrelated at
t
non-simultaneous lags. The (phi) 's and (theta) 's are k by k
i j
matrices of parameters. {(phi) }, for i=1,2,...,p, are called the
i
autoregressive (AR) parameter matrices, and {(theta) }, for
j
j=1,2,...,q, the moving average (MA) parameter matrices. The
parameters in the model are thus the p k by k (phi)-matrices, the
q k by k (theta)-matrices, the mean vector (mu) and the residual
error covariance matrix (Sigma). Let
[(phi) I 0 . . . 0]
[ 1 ]
[(phi) 0 I 0 . . 0]
[ 2 ]
[. . ]
A((phi))=[. . ]
[. . ]
[(phi) 0 . . . 0 I]
[ p-1 ]
[(phi) 0 . . . 0 0]
[ p ]pk*pk
and
[(theta) I 0 . . . 0]
[ 1 ]
[(theta) 0 I 0 . . 0]
[ 2 ]
[. . ]
B((theta))=[. . ]
[. . ]
[(theta) 0 . . . 0 I]
[ q-1 ]
[(theta) 0 . . . 0 0]
[ q ]qk*qk
where I denotes the k by k identity matrix.
The model (1) must be both stationary and invertible. The model
is said to be stationary if the eigenvalues of A((phi)) lie
inside the unit circle and invertible if the eigenvalues of
B((theta)) lie inside the unit circle.
At the user's request a new realisation of the time series may be
generated with less computation using only the information saved
in a reference vector from a previous call to G05HDF.
The routine returns a realisation of W ,W ,...,W . On a
1 2 n
successful exit, the recent history is updated and saved in the
array REF so that G05HDF may be called again to generate a
realisation of W ,W ,...,etc.
n+1 n+2
Parameters
g05hdf
Required Input Arguments:
ip integer
iq integer
par (:) real
qq (:,:) real
n integer
Optional Input Arguments: <Default>
mean (1) string 'z'
ref (:) real zeros(g05hdf13(k,ip,iq),1)
mode (1) string 's'
ifail integer -1
Output Arguments:
w (:,n) real
ref (:) real
ifail integer