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78
CHAPTER 14
SOLUTIONS TO PROBLEMS
14.1
First, for each
t
> 1, Var(
Δ
u
it
) = Var(
u
it
–
u
i,t
1
) = Var(
u
it
) + Var(
u
i,t
1
) =
2
2
u
σ
, where we use
the assumptions of no serial correlation in {
u
t
} and constant variance.
Next, we find the
covariance between
Δ
u
it
and
Δ
u
i,t+
1
.
Because these each have a zero mean, the covariance is
E(
Δ
u
it
⋅Δ
u
i,t+
1
) = E[(
u
it
–
u
i,t
1
)(
u
i,t+
1
–
u
it
)] = E(
u
it
u
i,t+
1
) – E(
2
it
u
) – E(
u
i,t
1
u
i,t+
1
) + E(
u
i,t
1
u
it
) =
−
E(
2
it
u
) =
2
u
−
because of the no serial correlation assumption.
Because the variance is constant
across
t
, by Problem 11.1, Corr(
Δ
u
it
,
Δ
u
i,t+
1
) = Cov(
Δ
u
it
,
Δ
u
i,t+
1
)/Var(
∆
u
it
) =
22
/(2
)
uu
−
=
−
.5.
14.3
(i) E(
e
it
) = E(
v
it
−
i
v
λ
) = E(
v
it
)
−
E(
i
v
) = 0 because E(
v
it
) = 0 for all
t
.
(ii) Var(
v
it
−
i
v
) = Var(
v
it
) +
2
Var(
i
v
)
−
2
⋅
Cov(
v
it
,
i
v
) =
2
v
+
2
E(
2
i
v
)
−
2
⋅
E(
v
it
i
v
).
Now,
2
2
E( )
vi
t
a
u
v
σσ
==
+
and E(
v
it
i
v
) =
1
1
()
T
it is
s
TE
v
v
−
=
∑
=
1
T
−
[
2
a
+
2
a
+
…
+ (
2
a
+
2
u
) +
…
+
2
a
] =
2
a
+
2
u
/
T
.
Therefore, E(
2
i
v
) =
1
1
T
it i
t
v
v
−
=
∑
=
2
a
+
2
u
/
T
.
Now, we can collect
terms:
Var(
v
it
−
i
v
)
=
2
(
/
)
2
(
/
)
au
TT
++
+
−
+
.
Now, it is convenient to write
= 1
−
/
η
γ
, where
≡
2
u
/
T
and
≡
2
a
+
2
u
/
T
.
Then
Var(
v
it
−
i
v
)
=
(
2
a
+
2
u
)
−
2
(
2
a
+
2
u
/
T
) +
2
(
2
a
+
2
u
/
T
)
= (
2
a
+
2
u
)
−
2(1
−
/
)
+ (1
−
/
)
2
= (
2
a
+
2
u
)
−
2
+ 2
⋅
+ (1
−
2
/
+
/
)
= (
2
a
+
2
u
)
−
2
+ 2
⋅
+ (1
−
2
/
+
/
)
= (
2
a
+
2
u
)
−
2
+ 2
⋅
+
−
2
⋅
+
= (
2
a
+
2
u
)
+
−
=
2
u
.
This is what we wanted to show.
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79
(iii) We must show that E(
e
it
e
is
) = 0 for
t
≠
s
.
Now E(
e
it
e
is
) = E[(
v
it
−
i
v
λ
)(
v
is
−
i
v
)] =
E(
v
it
v
is
)
−
E(
i
vv
is
)
−
E(
v
it
i
v
) +
2
E(
2
i
v
) =
2
a
σ
−
2
(
2
a
+
2
u
/
T
) +
2
E(
2
i
v
) =
2
a
−
2
(
2
a
+
2
u
/
T
) +
λ
2
(
2
a
+
2
u
/
T
).
The rest of the proof is very similar to part (ii):
E(
e
it
e
is
) =
2
a
−
2
λ
(
2
a
+
2
u
/
T
) +
λ
2
(
2
a
+
2
u
/
T
)
=
2
a
−
2(1
−
/
η
γ
)
+ (1
−
/
)
2
=
2
a
−
2
+ 2
⋅
+ (1
−
2
/
+
/
)
=
2
a
−
2
+ 2
⋅
+ (1
−
2
/
+
η
/
γ
)
γ
=
2
a
−
2
+ 2
⋅
+
−
2
⋅
+
=
2
a
+
−
= 0.
14.5
(i) For each student we have several measures of performance, typically three or four, the
number of classes taken by a student that have final exams.
When we specify an equation for
each standardized final exam score, the errors in the different equations for the same student are
certain to be correlated: students who have more (unobserved) ability tend to do better on all
tests.
(ii) An unobserved effects model is
score
sc
=
θ
c
+
β
1
atndrte
sc
+
2
major
sc
+
3
SAT
s
+
4
cumGPA
s
+
a
s
+
u
sc
,
where
a
s
is the unobserved student effect.
Because SAT score and cumulative GPA depend only
on the student, and not on the particular class he/she is taking, these do not have a
c
subscript.
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 Spring '11
 Suzuki
 Econometrics, Normal Distribution, Standard Error, Variance, Studentized residual

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