Problem Set 5
ECON 231W
Spring 2010
Suggested Solutions
1.
a)You cannot use OLS, because the equation under consideration is not of the form
αX
1
+
βX
2
+
ε,
where
ε
enters additively and the coefficients, which we estimate (in power 1), are multiplying the
explanatory variables.
b) You cannot compute
E
[
ε
i

L
i
K
i
] using the fact that
E
[ln
ε
i

L
i
K
i
] = 0, because expectation is
a linear operator and we are not given information about the whole conditional distribution of ln
ε
i
(but just its first moment). In this case, you can only conclude that
E
[ln
ε
i

L
i
K
i
] = 0
<
ln
E
[
ε
i

L
i
K
i
]
(as ln
x
is a concave function of
x
), which implies that
E
[
ε
i

L
i
K
i
]
>
1.
c) You can use take the log of the right hand side and left hand side of the equation to get:
ln
y
i
=
α
ln
L
i
+
β
ln
K
i
+ ln
ε
i
d) In the equation above let
e
y
≡
ln
y,
e
L
≡
ln
L,
e
K
≡
ln
K
,
e
ε
≡
ln
ε,
so we have a convenient linear
representation
e
y
i
=
α
e
L
i
+
β
e
K
i
+
e
ε
i
,
with
E
[
e
ε
i

L
i
, K
i
] =
E
[
e
ε
i

e
L
i
,
e
K
i
] = 0, which altogether satisfies OLS Assumption 1.
e) In economic terms,
α
is the elasticity of output with respect to labor labor input, and
β
is the
elasticity of output with respect to capital.
f) One way is to use a simple
t
test, where:
H
0
:
α
+
β
=
1
H
1
:
α
+
β
6
=
1
Where the standard error of
b
α
+
b
β
is
q
d
V ar
(
b
α
) +
d
V ar
(
b
β
) + 2
d
Cov
(
b
α,
b
β
)
.
An alternative method would
be to rewrite the equation as:
ln
y
i
=
α
ln
L
i
+
β
ln
K
i
+
u
i
=
α
ln
L
i
+
β
ln
L
i

β
ln
L
i
+
β
ln
K
i
+
u
i
=
(
α
+
β
) ln
L
i
+
β
(ln
K
i

ln
L
i
) +
u
i
=
γ
ln
L
i
+
β
(ln
K
i

ln
L
i
) +
u
i
And run a simple
t
test, where:
H
0
:
γ
=
1
H
1
:
γ
6
=
1
.
1
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2.
a.) Here the RHS variable is measured in levels, and the LHS in logs so the relationship is log
linear for
Fiveret
. Thus, the coefficient of .0016 on
Fiveret
implies that a 1% change (because the
units are in %) in
Fiveret
is expected to increase
Salbon
by .16% (a pretty small effect). Moreover,
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 Spring '10
 JoshuaKinsler
 Regression Analysis, Academic degree, Bachelor's degree, Master's degree, age change

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