Notes 4
Models for Stationary Time Series
:
General Linear Processes
A general linear process
{
Z
t
}
is one that can be represented as a weighted linear com
bination of the present and past terms of a white noise process:
Z
t
=
a
t
+
ψ
1
a
t

1
+
ψ
2
a
t

2
+
...
(1)
=
ψ
(
B
)
a
t
where
ψ
(
B
) = 1 +
ψ
1
B
+
ψ
2
B
2
+
...
,
B
is a ‘backward shift operator’ deﬁned by
BZ
t
=
Z
t

1
, B
k
Z
t
=
Z
t

k
.
If the RHS of (1) truly an inﬁnite series, then certain restrictions must be placed on
the
ψ
0
s
for the RHS to be mathematically meaningful. For our purposes, it suﬃces to
assume that
∞
X
i
=1
ψ
2
i
<
∞
as
V ar
(
Z
t
) = (
∑
∞
i
=0
ψ
2
i
)
σ
2
a
.
We should note that since
{
a
t
}
is unobservable, there is no loss of generality to assume
the coeﬃcient of
a
t
is 1 (we put
ψ
0
= 1).
E
(
Z
t
) = 0
, γ
k
=
σ
2
a
∞
X
i
=0
ψ
i
ψ
i
+
k
, k
≥
0
, ψ
0
= 1
.
A process with a nonzero mean
μ
may be obtained by adding
μ
to the RHS of (1) of the
general linear process. Since the mean does not aﬀect the covariance (or correlation)
structure of a process, we shall assume a zero mean until we begin ﬁtting the models to
the data.
Moving Average (MA) Process
In the case where only a ﬁnite number of
ψ
0
s are nonzero, we have what is called a
moving average process. In this case, we write
Z
t
=
a
t

θ
1
a
t

1

θ
2
a
t

2

...

θ
q
a
t

q
.
We call such a series a moving average of order
q
[Notation: MA(
q
)].
Here
ψ
0
= 1
, ψ
1
=

θ
1
, ,.
..,ψ
q
=

θ
q
, θ
j
= 0 for all
j > q
.
Example: MA(1).
Z
t
=
a
t

θa
t

1
1
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0
=
V ar
(
Z
t
) =
σ
2
a
(1 +
θ
2
)
γ
1
=
Cov
(
Z
t
,Z
t

1
) =

θσ
2
a
γ
k
= 0
, k
≥
2
ρ
1
=

θ
1 +
θ
2
ρ
k
= 0
, k
≥
2
.
Example: MA(2).
Z
t
=
a
t

θ
1
a
t

1

θ
2
a
t

2
γ
0
=
V ar
(
Z
t
) =
σ
2
a
(1 +
θ
2
1
+
θ
2
2
)
γ
1
=
Cov
(
Z
t
,Z
t

1
) =
Cov
(
a
t

θ
1
a
t

1

θ
2
a
t

2
,a
t

1

θ
1
a
t

2

θ
2
a
t

3
) = (

θ
1
+
θ
1
θ
2
)
σ
2
a
γ
2
=
Cov
(
Z
t
,Z
t

2
) =
Cov
(
a
t

θ
1
a
t

1

θ
2
a
t

2
,a
t

2

θ
1
a
t

3

θ
2
a
t

4
) =

θ
2
σ
2
a
γ
k
= 0
, k
≥
3
ρ
1
=

θ
1
+
θ
1
θ
2
1 +
θ
2
1
+
θ
2
2
ρ
2
=

θ
2
1 +
θ
2
1
+
θ
2
2
ρ
k
= 0
, k
≥
3
.
For MA(
q
),
ρ
k
=

θ
k
+
θ
1
θ
k
+1
+
θ
2
θ
k
+2
+
...
+
θ
q

k
θ
q
1+
θ
2
1
+
θ
2
2
+
...
+
θ
2
q
k
= 1
,
2
,...,q
;
0
k
≥
q
+ 1
.
Process Variance
γ
0
= (1 +
θ
2
1
+
...
+
θ
2
q
)
σ
2
a
.
Autoregressive (AR) Process
A
p
th
order autoregressive process
{
Z
t
}
satisﬁes the equation
Z
t
=
φ
1
Z
t

1
+
φ
2
Z
t

2
+
...
+
φ
p
Z
t

p
+
a
t
Notation: AR(
p
).
Example: AR(1).
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 Spring '08
 WU,KaHo

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