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58
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
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
)] =
00
0
/(
)
/
.
hh
γγ
⋅=
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
+ 2
⋅
0 =
2
z
+
2
e
.
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
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
/(
2
z
+
2
e
) > 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 timeconstant variable,
z
.
11.5
(i) The following graph gives the estimated lag distribution:
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59
lag
01
2
3
4
5
6
7
8
9
10
11
12
coefficient
0
.04
.08
.12
.16
By some margin, the largest effect is at the ninth lag, which says that a temporary increase in
wage inflation has its largest effect on price inflation nine months later.
The smallest effect is at
the twelfth lag, which hopefully indicates (but does not guarantee) that we have accounted for
enough lags of
gwage
in the FLD model.
(ii) Lags two, three, and twelve have
t
statistics less than two.
The other lags are statistically
significant at the 5% level against a twosided alternative.
(Assuming either that the CLM
assumptions hold for exact tests or Assumptions TS.1
′
through TS.5
′
hold for asymptotic tests.)
(iii) The estimated LRP is just the sum of the lag coefficients from zero through twelve:
1.172.
While this is greater than one, it is not much greater, and the difference from unity could
be due to sampling error.
(iv) The model underlying and the estimated equation can be written with intercept
α
0
and
lag coefficients
δ
0
,
1
,
…
,
12
.
Denote the LRP by
θ
0
=
0
+
1
+
…
+
12
.
Now, we can write
0
=
0
−
1
−
2
−
…
−
12
.
If we plug this into the FDL model we obtain (with
y
t
=
gprice
t
and
z
t
=
gwage
t
)
y
t
=
α
0
+ (
0
−
1
−
2
−
…
−
12
)
z
t
+
1
z
t
1
+
2
z
t
2
+
…
+
12
z
t
12
+
u
t
=
0
+
0
z
t
+
1
(
z
t
1
–
z
t
) +
2
(
z
t
2
–
z
t
) +
…
+
12
(
z
t
12
–
z
t
) +
u
t
.
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This note was uploaded on 09/11/2011 for the course ECONOMICS eco375 taught by Professor Suzuki during the Spring '11 term at University of Toronto Toronto.
 Spring '11
 Suzuki
 Econometrics

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