D.S.G. Pollock 267The usual recourse in the face of this problem is scrupulously to avoid the useof the cumulation operator∇−^1 (z)and to represent the integrated system only in
the form of (6.71). This is not a wholly adequate solution to the problem since, to
exploit the algebra of the operators, it is necessary to define the inverses of all of
the polynomial operators. An alternative solution is to constrain the disturbance
sequence to be absolutely summable, which appears to negate the assumption that
it is generated by a stationary stochastic process.
The proper recourse is to replace the process of indefinite summation by a definite
summation that depends upon supplying the system with initial conditions at
some adjacent points in time. To show what this entails, we may consider the
system of equations that is derived from (6.69) by settingt=0, 1,...,T−1. The
set ofTequations can be arrayed in a matrix format as follows:
⎡
⎢⎢
⎢⎢
⎢
⎢⎢
⎢⎢
⎢
⎣
y 0 y− 1 ... y−p
y 1 y 0 ... y 1 −p
..
...
...
...
.
yp yp− 1 ... y 0
..
...
...
.
yT− 1 yT− 2 ...yT−p− 1⎤
⎥⎥
⎥⎥
⎥
⎥⎥
⎥⎥
⎥
⎦⎡
⎢⎢
⎢⎢
⎣1
φ 1
..
.
φp⎤
⎥⎥
⎥⎥
⎦=⎡
⎢⎢
⎢⎢
⎢
⎢⎢
⎢⎢
⎢
⎣ε 0 ε− 1 ... ε−q
ε 1 ε 0 ... ε 1 −q
..
...
...
...
.
εq εq− 1 ... ε 0
..
...
...
.
εT− 1 εT− 2 ...εT−q− 1⎤
⎥⎥
⎥⎥
⎥
⎥⎥
⎥⎥
⎥
⎦⎡
⎢⎢
⎢⎢
⎣1
θ 1
..
.
θq⎤
⎥⎥
⎥⎥
⎦. (6.72)
Apart from the elementsy 0 ,y 1 ,...,yT− 1 andε 0 ,ε 1 ,...,εT− 1 , which fall within
the indicated period, these equations comprise the valuesy−p,...,y− 1 and
ε−q,...,ε− 1 , which are to be found in the top-right corners of the matrices, and
which constitute the initial conditions at the start-up time oft=0.
Each of the elements within this display can be associated with the power of
zthat is indicated by the value of its subscripted index. In that case, the system
can be represented by equation (6.70) with the constituent polynomials defined as
follows:
y(z)=y−pz−p+···+y 0 +y 1 z+···+yT− 1 zT−^1 ,ε(z)=ε−qz−q+···+ε 0 +ε 1 z+···+εT− 1 zT−^1 ,φ(z)= 1 +φ 1 z+···+φpzp andθ(z)= 1 +θ 1 z+···+θqzq.(6.73)This scheme applies regardless of the values of the roots of the polynomial opera-torsφ(z)andθ(z). Therefore, it can accommodate the case whereφ(z)=∇d(z)α(z),
which is that of equation (6.71). One of the virtues of this notation is that it is not
burdened by an explicit representation of the initial conditions. At a later stage, in
section 6.7, we shall need to represent the initial conditions explicitly.
A trend has only a tenuous existence within the context of a univariate ARIMA
model of the sort represented by equation (6.71). In such a model, it amounts
to nothing more that the accumulation of the fluctuations that are created by