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H19
HW10 (due Apr. 10)
Problem 1
Give 1step and 2step volatility forecasts for the ARCH(1) model with
01
1.5,
0.9
α
==
in terms of the known values of
2
t
X
and
2
t
σ
.
Problem 2
Give a 3step volatility forecast for an ARCH(2) model with
2
1,
0.5,
0.2
αα
=
in terms of the known values of
2
t
X
and
2
t
.
Problem 3
Give 1step and 2step volatility forecasts for the GARCH (1,1) model with
011
1.5,
0.6,
0.3
αβ
===
in terms of the known values of
2
t
X
and
2
t
.
__________________________________________________________________________________________________
A
generalized
autoregressive conditional heteroscedastic (GARCH) model
with order
and
is defined as
(1
)
≥
p
(0
)
≥
q
σε
=
tt
X
t
2
and
22
2
2
1
1
1
β
−−
−
=
+
+⋅⋅⋅+
+
p
t
p
t
XX
−
q
t
q
,
(1)
where
0
≥
i
and
0
≥
j
are constants,
{}~
I
ID
(
0
,
1
)
ε
t
,
and
t
is independent of
{,
for all
t
.
1
−
≥
tk
Xk
}
A stochastic process
{}
t
X
defined by the equations above is called a GARCH
(,)
p q
process.
Formally,
2
t
X
follows an ARMA
(,
∨
)
p qq
model, where
max{ , }
p
qp
∨=
q
−
:
2
2
00
11
1
1
()
∨
−
=
=
=+
+
+
+
+
−
∑∑
∑
∑
pq
p
q
q
t
i ti
j t j
t
i
i
ti
t
jt j
ij
i
j
X
Xe
X
e
e
,
(2)
where
0
++
pj
qj
for
, and
.
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 Spring '09
 gUR

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