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Sum
(Integration)
Inverse AR
MA
White
n oise
Second
Filtering
Third
Filtering
First Filtering
a t
e t
w t
x t
Figure 2.2. Block diagram of an ARIMA model
From the above it follows that the general ARIMA process may be generated from
white noise a t by means of three filtering operation, as shown by the block diagram
in Figure 2.2, where the first filter has the input a t , the transfer function T( B ) , and
output
e
T
()
Ba
,
t
t
where
2
q
T
(
B
)
(1
T
B
T
B
...
T
B
)
1
2
q
is the moving-average operator.
Table 2.1. Summary of properties of AR, MA and ARMA processes
Sl. No.
Description
AR
MA
ARMA
1.
Model in
t
t
I
Bx
T
1
T
1
a
Bx
a
BBxa
I
t
t
t
t
terms of x
2.
Model in
terms of a
I
1
T
B
I
1
x
a
x
a
x
B
T
Ba
t
t
t
t
t
t
3.
\ = weights
Infinite series
Finite series
Infinite series
4.
Stationary
condition
Roots of
0
Always
stationary
Roots of 0
I
B
I
lie
outside
unit circle
B
lie outside unit
circle
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