matrix of a multivariate autoregressive moving average model, and obtains the closed ... Some key words: Autoregressive moving average process; Covariance ...
Biometrika (1997), 84,4, pp. 957-964 Printed in Great Britain
On the exact likelihood function of a multivariate autoregressive moving average model BY CHUNSHENG MA School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia SUMMARY
This paper derives the explicit expressions for the determinant and exact inverse of the covariance matrix of a multivariate autoregressive moving average model, and obtains the closed form of the exact likelihood function. Some key words: Autoregressive moving average process; Covariance matrix; Likelihood function. 1. INTRODUCTION
Assume that {X,} is an m-dimensional vector-valued time series generated by the zero-mean autoregressive moving average ARMA (p, q) process X ( -(D 1 X r _ 1 -...-(D p X ( _ p = Z ( - 0 1 Z r _ 1 - . . . - 0 , Z ( _ , ,
(11)
where l,...,p, © j , . . . , 0 , are real mx m parameter matrices, and {Z,} constitutes a sequence of independent m-dimensional normal random vectors each having mean zero and positive definite covariance matrix Z. Its stationarity and invertibility are determined by |/ m -O> 1 z=...-(D p z''|*0,
^-©.z-.-.-O^I+O,
for all complex numbers z such that |z| ^ 1, where Im is the identity mxm matrix and j^4| is the determinant of A. Problems of statistical inference for a multivariate ARMA (p, q) process often involve calculating the inverse of the covariance matrix of a set of successive observations, Xl,X2,...,Xn, say. The corresponding likelihood function of {Ox,..., 0, (3-2) and (3-3) are applicable provided that Z and X* are uncorrelated. One obtains
Under the stationary condition, (3-4) yields an equation
from which Q* can be expressed in terms of £ and the autoregressive parameters. Consequently, (3-2) and (3-3) simplify to {H,
H2) \
o
/„.
To derive the Cholesky decomposition of Q 1, it suffices to factorise Q* and E in the form C* = Co (Co ) T . £ = Eo^o. where Co ar>d Eo are lower triangular matrices. Qearly, such a decomposition is independent of the sample size n. As a result, C 1 = £ T £, where IS)"1 0 0 7._r
\flmp
0\
(
(Co*)"1
0
A similar result is given by Luceno (1994). However, such a factorisation is not necessary for deriving the closed form, see (4-2) below, of the exact likelihood function. Case (ii): MA (q). For a multivariate MA (q) model, H = I^,, F = 0, Z and Z* are uncorrelated; AQ is positive definite and A n = Do(7,® £)£>J is nonnegative definite. So |AJ > 0, and
In the particular case where q = 1, a direct application of the formula (2-2) gives Q-1 = A~l-MTAllM,
(3-5)
962
CHUNSHENG M A
where QTlB
Q"rl)TBm),
22..
-"22
B22
n —l.Fi — 1
n-2
©1
Alternatively, from (2-4) we obtain where the blocks of A are zero except for the first row with /Bn =A
©IB22 0
\
(®l
0
Such a representation holds obviously for an MA(q) model. Recently, Haddad (1995) has derived a similar result for the univariate MA (1) model. Case (iii): ARMA(1, 1). In this case Q~* can be written as Q-1=HTA-1H-HTMTAnMH,
(36)
l
where A~ and M are the same as in (3-5) but A u is replaced by
where An = 0 ^ 0 1 - ^{Q^yej
- ^ c * * ^ ! + ^i
or, if the model is stationary, A n = ^Q*]' — Z. Case (iv): ARMA (p, q) (p ^ q). The idea of deriving the explicit formulae (3-5) and (3-6) can be used to obtain an expression for Q" 1 of an ARMA (p, q) (p ^ q) model. To see this, note that there exist integers n^ and nt (0^nl< q) such that n = n^q + nlt 0
\
C= \ 0
where Coo (^oo) is the ^
Coo/
i,mx qm) matrix having the similar form as Co (Do), and /Co1
A,
0 \ 1
c^
5n
n ^00
^-00 '
Miscellanea 1
where Do =
963
(i = 1,..., no). Equation (2-2) implies that B22D0
H,
where the (i, j)th block of the symmetric matrix A
l
(3-7)
is
n-f
Y (5T)*(/ ®E~ X )5*
*=o
(i =
and 11)-\
A u = D0(Iq®i:)DT0-F0(Q**)T
DT0-D0Q**F
l
4. THE CLOSED FORM OF THE EXACT LIKELIHOOD
Substituting (3-2) and (3-3) in (1-2) we obtain the exact likelihood function as (4-1) Various approximate likelihood functions may be obtained from the closed form (41). For instance, if At is ignored, (41) reduces to an approximate likelihood function that does not depend on the preperiod values. Case (i): AR (p). For an AR (p) model, under the stationary condition (4-1) gives
(27i)-*"ui|zr*
