CHAPTER 11
SOLUTIONS TO PROBLEMS
11.1
Because of covariance stationarity,
0
γ
= Var(
x
t
) does not depend on
t
, so sd(
x
t+h
) =
0
γ
for
any
h
≥
0.
By definition, Corr(
x
t
,x
t
+
h
) = Cov(
x
t
,x
t+h
)/[sd(
x
t
)
⋅
sd(
x
t+h
)] =
0
0
/(
)
/
.
h
h
0
γ
γ
γ
γ
γ
⋅
=
11.3
(i) E(
y
t
) = E(
z
+
e
t
) = E(
z
) + E(
e
t
) = 0.
Var(
y
t
) = Var(
z
+
e
t
) = Var(
z
) + Var(
e
t
) +
2Cov(
z
,
e
t
) =
2
z
σ
+
2
e
σ
+ 2
0 =
⋅
2
z
σ
+
2
e
σ
.
Neither of these depends on
t
.
(ii) We assume
h
> 0; when
h
= 0 we obtain Var(
y
t
).
Then Cov(
y
t
,
y
t+h
) = E(
y
t
y
t+h
) = E[(
z
+
e
t
)(
z
+
e
t+h
)] = E(
z
2
) + E(
ze
t+h
) + E(
e
t
z
) + E(
e
t
e
t+h
) = E(
z
2
) =
2
z
σ
because {
e
t
} is an uncorrelated
sequence (it is an independent sequence and
z
is uncorrelated with
e
t
for all
t
.
From part (i) we
know that E(
y
t
) and Var(
y
t
) do not depend on
t
and we have shown that Cov(
y
t
,
y
t+h
) depends on
neither
t
nor
h
.
Therefore, {
y
t
} is covariance stationary.
(iii) From Problem 11.1 and parts (i) and (ii), Corr(
y
t
,
y
t+h
) = Cov(
y
t
,
y
t+h
)/Var(
y
t
) =
2
z
σ
/(
2
z
σ
+
2
e
σ
) > 0.
(iv) No.
The correlation between
y
t
and
y
t+h
is the same positive value obtained in part (iii)
now matter how large is
h
.
In other words, no matter how far apart
y
t
and
y
t+h
are, their
correlation is always the same.
Of course, the persistent correlation across time is due to the
presence of the time-constant variable,
z
.
11.5
(i) The following graph gives the estimated lag distribution:
58

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