H18
Problem 1
Consider the following model:
13
4
0.8
0.125
0.1
tt
t
t
t
X
XX
X
−−
−
=+
−+
Z
.
(a)
Is the model stationary?
(b)
Find a statespace representation of the model.
Problem 2
Find a statespace representation of the
MA(2)
process
11 2 2
t
t
X
ZZ
Z
θ
=
++
.
Models for Changing Variance
.
The model is called
homoscedastic
if the variance is constant, and
heteroscedastic
when it is not.
Example 1
Annual numbers of lynx trapped in the Mackenzie River district of Canada.
Example 2
Dow Jones Industrial Index.
An
autoregressive conditional heteroscedastic (ARCH) model
with order
is defined as
(1
)
≥
p
X
t
σε
=
and
22
01
1
p
2
t
p
X
X
σα
α
−
−
+
⋅
⋅
⋅
+
,
where
0
≥
i
are constants,
{}~
I
ID
(
0
,
1
)
ε
t
, and
t
is independent of
{,
for all
t
.
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 Spring '09
 gUR
 Stochastic process, Autocorrelation, Stationary process, Autoregressive moving average model, Mathematical terminology, Necessary and sufficient condition

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