This preview shows pages 1–2. Sign up to view the full content.
Practice question 2:
Consider the ARIMA(1,1,0) model
)
,
0
(
~
,
)
9
.
0
1
)(
1
(
2
σ
NID
a
a
X
B
B
t
t
t
=
+

.
The most recent 8 observations for 1989 to 1996 were
)
3
.
6
,
2
.
7
,
9
.
4
,
3
.
4
,
2
.
2
,
5
.
1
,
1
.
0
,
0
(
)
,
,
(
96
89







=
X
X
.
(a)
Write out the recursive formula for the
t
X
values.
l
t
l
t
l
t
l
t
t
t
t
t
t
t
t
t
a
X
X
X
a
X
X
X
a
X
B
B
a
X
B
B
+
+

+

+


+
⋅
+
⋅
=
⇒
+
⋅
+
⋅
=
⇒
=


⇒
=
+

2
1
2
1
2
9
.
0
1
.
0
9
.
0
1
.
0
)
9
.
0
1
.
0
1
(
)
9
.
0
1
)(
1
(
3
),
2
(
ˆ
9
.
0
)
1
(
ˆ
1
.
0
)
(
ˆ
)
1
(
ˆ
9
.
0
)
2
(
ˆ
1
.
0
)
3
(
ˆ
9
.
0
)
1
(
ˆ
1
.
0
)
2
(
ˆ
9
.
0
1
.
0
)
1
(
ˆ
1
≥

⋅
+

⋅
=
⋅
+
⋅
=
⋅
+
⋅
=
⋅
+
⋅
=

l
l
X
l
X
l
X
X
X
X
X
X
X
X
X
X
t
t
t
t
t
t
t
t
t
t
t
t
(b) Derive the formulas for predictions for 1997 to 1999 in terms of previously
observed values. (These may be expressed in terms of other predictions, as
long as you describe how to calculate each term before you use it another
formula.)
0371
.
7
)
3
(
ˆ
]
[
381
.
6
)
2
(
ˆ
]
[
11
.
7
)
1
(
ˆ
]
[
96
99
96
96
98
96
96
97
96

=
=

=
=

=
=
X
X
E
X
X
E
X
X
E
(c)
Let
X
be the mean value of this series in the 90’s (i.e., from 1990 to 1999).
Use the values above and your formulas to calculate the estimate
X
ˆ
of
X
.
(You should give this estimate both as a formula and numerically.)
Assume
that all earlier values of the series are zero if you need them in your
predictions.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/27/2009 for the course STA STA457 taught by Professor Lin during the Winter '08 term at University of Toronto Toronto.
 Winter '08
 Lin
 Forecasting

Click to edit the document details