Information Technology Reference
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where
I
is the
stationary autoregressive operator
and
M
is the
generalized
autoregressive operator
. One or more of the zeros of the polynomial
M
(
i.e
. one
or more of the roots of the equation
M
) is unity.
The general form of a model to represent the homogeneous nonstationary
behaviour of the time series is given by
() 0
M
()()
Bxt
{
I
( (1
B
B xt
) ()
d
or,
M
()()
Bxt
T
() .
t
Ba
Defining now
wt
()
(1
B
)
d
xt
()
,
we get
M
()()
Bwt
T
() .
t
Ba
Introducing the operator
(1
B
)
the last equation becomes
d
wt
()
xt
().
where,
{
x t
()
xt
()
xt
(
1)
(1
Bxt
) ().
The homogeneous nonstationary behaviour can therefore be represented by a
model which calls for the
d
th difference of the process to be stationary. In practice,
it is mostly
d
= 0 or 1 but not greater than 2.
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