a. Find the autocovariance and autocorrelation functions for this process when
θ
.
8.
b. Compute the variance of the sample mean
(X
1
+
X
2
+
X
3
+
X
4
)/
4 when
θ
.
8.
c. Repeat (b) when
θ
−
.
8 and compare your answer with the result obtained
in (b).
1.6.
Let
{
X
t
}
be the AR(1) process defined in Example 1.4.5.
a. Compute the variance of the sample mean
(X
1
+
X
2
+
X
3
+
X
4
)/
4 when
φ
.
9 and
σ
2
1.
b. Repeat (a) when
φ
−
.
9 and compare your answer with the result obtained
in (a).
1.7.
If
{
X
t
}
and
{
Y
t
}
are uncorrelated stationary sequences, i.e., if
X
r
and
Y
s
are
uncorrelated for every
r
and
s
, show that
{
X
t
+
Y
t
}
is stationary with autoco
variance function equal to the sum of the autocovariance functions of
{
X
t
}
and
{
Y
t
}
.
42
Chapter 1
Introduction
1.8.
Let
{
Z
t
}
be IID N
(
0
,
1
)
noise and define
X
t
Z
t
,
if
t
is even
,
(Z
2
t
−
1
−
1
)/
√
2
,
if
t
is odd
.
a. Show that
{
X
t
}
is WN
(
0
,
1
)
but not iid
(
0
,
1
)
noise.
b. Find
E(X
n
+
1

X
1
, . . . , X
n
)
for
n
odd and
n
even and compare the results.
1.9.
Let
{
x
1
, . . . , x
n
}
be observed values of a time series at times 1
, . . . , n
, and let
ˆ
ρ(h)
be the sample ACF at lag
h
as in Definition 1.4.4.
a. If
x
t
a
+
bt
, where
a
and
b
are constants and
b
0, show that for each
fixed
h
≥
1,
ˆ
ρ(h)
→
1 as
n
→ ∞
.
b. If
x
t
c
cos
(ωt)
, where
c
and
ω
are constants (
c
0 and
ω
∈
(
−
π, π
]),
show that for each fixed
h
,
ˆ
ρ(h)
→
cos
(ωh)
as
n
→ ∞
.
1.10.
If
m
t
∑
p
k
0
c
k
t
k
,
t
0
,
±
1
, . . . ,
show that
∇
m
t
is a polynomial of degree
p
−
1 in
t
and hence that
∇
p
+
1
m
t
0.
1.11.
Consider the simple movingaverage filter with weights
a
j
(
2
q
+
1
)
−
1
,
−
q
≤
j
≤
q
.
a. If
m
t
c
0
+
c
1
t
, show that
∑
q
j
−
q
a
j
m
t
−
j
m
t
.
b. If
Z
t
, t
0
,
±
1
,
±
2
, . . . ,
are independent random variables with mean 0 and
variance
σ
2
, show that the moving average
A
t
∑
q
j
−
q
a
j
Z
t
−
j
is “small”
for large
q
in the sense that
EA
t
0 and Var
(A
t
)
σ
2
/(
2
q
+
1
)
.
1.12.
a. Show that a linear filter
{
a
j
}
passes an arbitrary polynomial of degree
k
without distortion, i.e., that
m
t
j
a
j
m
t
−
j
for all
k
thdegree polynomials
m
t
c
0
+
c
1
t
+ · · · +
c
k
t
k
, if and only if
j
a
j
1
and
j
j
r
a
j
0
,
for
r
1
, . . . , k.
b. Deduce that the Spencer 15point movingaverage filter
{
a
j
}
defined by
(1.5.6) passes arbitrary thirddegree polynomial trends without distortion.
Problems
43
1.13.
Find a filter of the form 1
+
αB
+
βB
2
+
γ B
3
(i.e., find
α
,
β
, and
γ
) that
passes linear trends without distortion and that eliminates arbitrary seasonal
components of period 2.
1.14.
Show that the filter with coefficients [
a
−
2
, a
−
1
, a
0
, a
1
, a
2
]
1
9
[
−
1
,
4
,
3
,
4
,
−
1]
passes thirddegree polynomials and eliminates seasonal components with pe
riod 3.
1.15.
Let
{
Y
t
}
be a stationary process with mean zero and let
a
and
b
be constants.
a. If
X
t
a
+
bt
+
s
t
+
Y
t
, where
s
t
is a seasonal component with period
12, show that
∇∇
12
X
t
(
1
−
B)(
1
−
B
12
)X
t
is stationary and express its
autocovariance function in terms of that of
{
Y
t
}
.
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 Fall '14
 Statistics, The Land, Stationary process, Time series analysis, Bartlett Press