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Economics 326
Methods of Empirical Research in Economics
Lecture 14: Hypothesis testing in the multiple
regression model, Part 2
Vadim Marmer
University of British Columbia
May 5, 2010
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I
Consider the model:
ln
(
Wage
i
) =
β
0
+
β
1
Experience
i
+
β
2
Experience
2
i
+
+
β
3
PrevExperience
i
+
β
4
PrevExperience
2
i
+
β
5
Education
i
+
U
i
,
where
Experience
is the experience at current job, and
PrevExperience
is the previous experience.
I
Suppose that we want to test the null hypothesis that, after
controlling for the experience at current job and education,
the previous experience has no e/ect on wage:
H
0
:
β
3
=
0
,
β
4
=
0
.
I
We have two restrictions on the model parameters.
I
The alternative hypothesis is that at least one of the
coe¢ cients,
β
3
or
β
4
,
is di/erent from zero:
H
1
:
β
3
6
=
0 or
β
4
6
=
0
.
1±23
t
statistics and multiple restrictions
I
Let
T
3
and
T
4
be the
t
statistics associated with the
coe¢ cients of
PrevExperience
and
PrevExperience
2
:
T
3
=
ˆ
β
3
SE
ˆ
β
3
±
and
T
4
=
ˆ
β
4
SE
ˆ
β
4
±
.
I
We can use
T
3
and
T
4
β
3
and
β
4
separately by using two separate size
α
tests:
I
Reject
H
0
,
3
:
β
3
=
0 in favor of
H
1
,
3
:
β
3
6
=
0 when
j
T
3
j
>
t
n
k
1
,
1
α
/
2
.
I
Reject
H
0
,
4
:
β
4
=
0 in favor of
H
1
,
4
:
β
4
6
=
0 when
j
T
4
j
>
t
n
k
1
,
1
α
/
2
.
2/23
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statistics and multiple restrictions
I
Rejecting
H
0
:
β
3
=
0
,
β
4
=
0 in favor of
H
1
:
β
3
6
=
0 or
β
4
6
=
at level
α
, i.e. when
j
T
3
j
>
t
n
k
1
,
1
α
/
2
or
j
T
4
j
>
t
n
k
1
,
1
α
/
2
,
is not a size
α
test!
I
Recall that if
A
and
B
are two sets then
(
A
\
B
)
±
A
and
therefore
P
(
A
\
B
)
²
P
(
A
)
.
I
When
β
3
=
β
4
=
0 :
P
(
Reject
H
0
,
3
or
H
0
,
4
) =
=
P
(
j
T
3
j
>
t
n
k
1
,
1
α
/
2
or
j
T
4
j
>
t
n
k
1
,
1
α
/
2
)
=
P
(
j
T
3
j
>
t
n
k
1
,
1
α
/
2
) +
P
(
j
T
4
j
>
t
n
k
1
,
1
α
/
2
)
P
(
j
T
3
j
>
t
n
k
1
,
1
α
/
2
and
j
T
4
j
>
t
n
k
1
,
1
α
/
2
)
=
α
+
α
P
(
j
T
3
j
>
t
n
k
1
,
1
α
/
2
and
j
T
4
j
>
t
n
k
1
,
1
α
/
2
)
³
α
.
3/23
I
Consider the model
Y
i
=
β
0
+
β
1
X
1
,
i
+
. . .
+
β
q
X
q
,
i
+
β
q
+
1
X
q
+
1
,
i
+
. . .
+
β
k
X
k
,
i
+
U
i
.
q
regressors have
no e/ect on
Y
(after controlling for other regressors).
I
The null hypothesis has
q
exclusion restrictions:
H
0
:
β
1
=
0
,
β
2
=
0
,
. . .
,
β
q
=
0
.
I
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This note was uploaded on 03/24/2012 for the course ECON 326 taught by Professor Whisler during the Spring '10 term at The University of British Columbia.
 Spring '10
 whisler
 Economics

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