H3
For
MA(
q
)
processes,
2
0
,0
,
()
0,
.
σθ
θ
γ
−
+
=
⎧
≤
≤
⎪
=
⎨
⎪>
⎩
∑
qh
jjh
j
hq
h
0
22
1
,
1
0,
.
θθ
ρ
−
+
=
⎧
⎪
≤
≤
⎪
=
⎨
+
+⋅⋅⋅+
⎪
>
⎪
⎩
∑
j
q
h
The ACF of a MA(
q
) process “cuts off” after the point
q
.
This is a benchmark property for MA processes.
AR(1):
1
0
i
ttt
t
i
i
X
XZ
Z
φφ
∞
−−
=
=+
=
∑
.
2
4
2
2
0
()0
,V
a
r
(
1
)
1
σ
σφ
φ
.
ϕ
∞
=
==
+
+
+
=
=
−
∑
L
i
tt
i
EX
X
/
(
1
)
hh
X
h
2
φσ
=−
=
, provided
1
<
( )
,
0,1, 2, .
...
h
ρφ
⇒=
=
.
Example 1
1
0.9
t
X
X
−
Z
t
and
1
0.9
X
−
=
−+
.
A process
{ }
t
X
is said to be
invertible
if the random disturbance at time
t
can be expressed
as a convergent sum of present and past values of the process in the form
0
tj
j
t
j
Z
X
π
∞
−
=
=
∑
,
where

j
< ∞
∑
.
This effectively means that the process
can be rewritten in the form of an AR process
whose coefficients form a convergent sum.
By using the backward shift operator
1
,,
j
j
B
BX
X
B X
X
−
−
,
MA(
q
)
,
11
t
q
t
q
X
ZZ
Z
−
=
+
−
, can be presented as
X
BZ
=
.
where
1
() 1
q
q
B
B
θθθ
+
⋅
⋅
⋅
+
B
is a polynomial of order
q
in
B.
MA(
q
) process is
invertible
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 Spring '09
 gUR
 zt, 0 j, convergent sum, 0 j

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