AMS316 Quiz #3
1. Compute the ACF
ρ
(
k
) (
k
≥
0) of the AR(2) process 8
X
t
= 2
X
t

1
+
X
t

2
+
Z
t
, in
which
Z
t
are i.i.d. standard normal random variables.
2. Consider the stationary AR(2) process
x
t
=
α
1
x
t

1
+
α
2
x
t

2
+
z
t
, in which
z
t
are i.i.d.
normal random variables with mean 0 and variance
σ
2
. The observations we have are
x
1
, . . . , x
n
.
(a) What is the 1step ahead forecast at the forcast origin
x
n
, i.e.,
x
n
(1)? What is
the forcast error and its variance?
(b) What is the 2step ahead forecast at the forcast origin
x
n
, i.e.,
x
n
(2)? What is
the forcast error and its variance?
(c) What is the
k
step ahead forecast at the forcast origin
x
n
, i.e.,
x
n
(
k
), for
k
≥
3?
(It is OK to write down the recursive formula for
x
n
(
k
))
3. Consider the stationary and invertible ARMA(1,1) process
x
t

αx
t

1
=
z
t

θz
t

1
,
in which
z
t
are i.i.d. normal random variables with mean 0 and variance
σ
2
. The
observations we have are
x
1
, . . . , x
n
.
(a) What is the 1step ahead forecast at the forcast origin
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This note was uploaded on 08/08/2010 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.
 Fall '09

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