PSTAT 174 / 274  Midterm
Consider GDP from 1983 1Q to 2002 4Q. Figure 1 shows the linear trend over time and you
plotted the deterended time series with differentiation.
GDP
Time
Billion $
0
20
40
60
80
4000
6000
8000
10000
(a)
Detrended Time Series with Differentiation
Time
Billion $
0
20
40
60
80
0
50
100
150
200
(b)
Figure 1:
GDP data from 1983 1Q to 2002 4Q
1. (30 points) You can consider the following model,
x
t
=
β
1
+
β
2
t
+
w
t
,
where
β
1
and
β
2
are known constants and
w
t
is a white noise process with variance
σ
2
.
Prove that the first difference series
∇
x
t
=
x
t

x
t

1
is stationary by finding its mean and
autocovariance function.
2. (30 points) Consider the following AR(1) time series model
X
t
= 0
.
4
X
t

1
+
w
t
, w
t
∼
i.i.d N
(0
,
1225)
.
(a) Find and sketch the ACF,
ρ
(
h
) .
(b) Find and sketch the PACF,
φ
hh
.
1
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3. (10 points) You used differentiation to remove the linear trend and plotted ACF/PACF.
Identify the best linear time series model that you want to fit the detrended time series.
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 Spring '08
 MACKGALLOWAY
 Regression Analysis, Harshad number, Time series analysis, Trend estimation

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