5/10/2010
1
Dummy Variables in MR  VIII
•
We also may be interested in the effects of
multiple sets
of categorical
variables, e.g. the effects of both the sex of an individual,
and
their ethnicity.
In this case, there needs to be an “excluded” group for each set of categories,
e.g.
–
Ethnicity: let’s choose “Other” as the excluded group
•
Hispanic
i
= 1 if observation
i
is Hispanic, Hispanic
i
= 0 otherwise
•
Black
i
= 1 if observation
i
is Black, Black
i
= 0 otherwise
–
Sex: let’s choose “Male” as the excluded group
•
Female
i
= 1 if observation
i
is Female, Female
i
= 0 otherwise
•
What do these regression results say about average wages for Hispanic
Females, Black Males, Other Males, etc………?
circumflexnosp4char
Wage = 17.40
6.08Hispanic
4.59Black
2.90Female
(1.02)
(1.10)
(1.04)
(0.97)
i
i
i
i



Hypothesis Testing in MR
•
There are two types of hypothesis tests we can do in multiple
regression.
–
1) Hypothesis tests involving a
single coefficient
, e.g. a test
whether
β
5
=1, or a test whether
β
3
=0.
–
2) Hypothesis tests involving
multiple coefficients
, e.g. a test
whether
β
5
=2*
β
3
, or a test whether
β
1
=
β
2
=
β
4
=0.
These are
sometimes called
joint
hypothesis tests, because they test
conditions on multiple coefficients jointly, or together.
•
Tests of the form 1) can be done using
t
STAT
’s, just like in basic
regression analysis. As discussed last week, these are done
exactly like in simple regression.
•
Tests of the form 2) will be done with a
new statistic
, the F
statistic, or
F
STAT
.
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2
Hypothesis Tests for a Single
Coefficient  I
•
One new issue regarding tests for a single coefficient:
When using dummy variables,
remember that we had to choose an excluded group.
In our PS2 example we had:
•
Regression A (with “Other” as the excluded group)
•
Regression B (with “Hispanic” as the excluded group)
•
Again, these regressions are really
exactly
the same (they both say average wages for
Others is 16.08, average wages for Blacks is 11.57, and average wages for Hispanics
is 10.19)
•
But because the variables are coded differently, the coefficients measure different
aspects of the relationship, so the
t
STAT
’s on the coefficients test different things.
circumflexnosp4char
Wage = 16.08
5.89Hispanic
4.51Black
(0.69)
(1.09)
(1.84)
i
i
i


circumflexnosp4char
Wage
= 10.19 + 1.38Black
5.89Other
(0.85)
(1.94)
(1.09)
i
i
i
+
Hypothesis Tests for a Single
Coefficient  II
•
For example, in Regression A, the coefficient on the variable Black
i
measures
the difference between average wages for Blacks and average wages for
Others. (since Other is the excluded group).
Hence the
t
STAT
for the
hypothesis test that this coefficient equals 0 tests the hypothesis that Blacks
and Others have equal average wages.
•
On the other hand, in Regression B, the coefficient on the variable Black
i
measures the difference between average wages for Blacks and average
wages for Hispanics. (since Hispanics is now the excluded group).
Hence the
t
STAT
for the hypothesis test that this coefficient equals 0 tests the hypothesis
that Blacks and Hispanics have equal average wages.
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 Spring '07
 SandraBlack
 Statistics, Econometrics, Null hypothesis, Statistical hypothesis testing, Statistical power, fstat

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