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AMS316 HW6
(Due Dec 12, 2011)
1. Consider the AR(1) process,
X
t
=
μ
+
αX
t

1
+
Z
t
,
where
Z
t
are i.i.d.
standard normal random variables. Derive the least square estimates
for
μ
and
α
by minimizing
S
(
μ,α
) =
n
X
t
=1
(
X
t

μ

αX
t

1
)
2
.
2. For the MA(1) model given by
X
t
=
Z
t
+
θZ
t

1
and observations
X
1
,...,X
N
, show that the 1step ahead forecast
b
X
N
(1) =
θZ
N
and
that the
h
step ahead forecast
b
X
N
(
h
) = 0 for
h
= 2
,
3
,...
.
3. For the AR(1) model given by
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