Extensions

1

• Extensions
• of the Multiple
• Regression Model

2
Topics for This Chapter

• 1. Intercept Dummy Variables
• 2. Slope Dummy Variables
• 3. Different Intercepts Slopes
• 4. Testing Qualitative Effects
• 5. Are Two Regressions Equal?
• 6. Interaction Effects

3
Dummy variables

• Dummy variables, often called binary or
  dichotomous variables, are explanatory variables
  that only take two values, usually 0 and 1.
• These simple variables are a very powerful tool
  for capturing qualitative characteristics of
  individuals, such as gender, race, geographic
  region of residence.
• In general, we use dummy variables to describe
  any event that has only two possible outcomes.

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Intercept Dummy Variables
Dummy variables are binary (0,1)
yt ?1 ?2Xt ?3Dt et
yt speed of car in miles per hour
Xt age of car in years
Dt 1 if red car, Dt 0 otherwise.
H0 ?3 0
Police red cars travel faster.
H1 ?3 gt 0
5
yt ?1 ?2xt ?3Dt et
red cars yt (?1 ?3) ?2xt et
other cars yt ?1 ?2xt et
yt
?1 ?3
?2
?1
red cars
miles per hour
?2
other cars
0
Xt
age in years
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Slope Dummy Variables
yt ?1 ?2Xt ?3DtXt et
Stock portfolio Dt 1 Bond portfolio Dt 0
yt
yt ?1 (?2 ?3)Xt et
stocks
Value of portfolio
?2 ?3
bonds
?2
?1
yt ?1 ?2Xt et
?1 initial investment
0
Xt
years
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Different Intercepts Slopes
yt ?1 ?2Xt ?3Dt ?4DtXt et
Miracle seed Dt 1 regular seed Dt 0
yt
yt (?1 ?3) (?2 ?4)Xt et
harvest weight of corn
Miracle
?2 ?4
yt ?1 ?2Xt et
regular
?1 ?3
?2
?1
Xt
rainfall
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yt ?1 ?2 Xt ?3 Dt et
For men? Dt 1. For women? Dt 0.
yt
Men
yt (?1 ?3) ?2 Xt et
wage rate
Women
?2
yt ?1 ?2 Xt et
.
Testing for discrimination in starting wage
?2
?1 ?3
.
?1
0
years of experience
Xt
9
yt ?1 ?5 Xt ?6 Dt Xt et
For men Dt 1. For women Dt 0.
yt
yt ?1 (?5 ??6 )Xt et
Men
wage rate
?5 ??6
Women
yt ?1 ?5 Xt et
?5
Men and women have the same starting wage, ?1
, but their wage rates increase at different
rates (different ?6 ). ?6 gt ?? means that
mens wage rates are increasing faster than
women's wage rates.
?1
0
years of experience
Xt
10
yt ?1 ?2 Xt ?3 Dt ?4 Dt Xt et
An Ineffective Affirmative Action Plan
women are started at a higher wage.
yt
yt (?1 ?3) (?2 ?4) Xt et
Men
wage rate
?2 ?4
Women
yt ?1 ?2 Xt et
?2
Women are given a higher starting wage, ?1 ,
while men get the lower starting wage, ?1 ?3
, (?3 lt 0 ). But, men get a faster rate of
increase in their wages, ?2 ?4 , which is
higher than the rate of increase for women, ?2 ,
(since ?4 gt 0 ).
?1
?1 ?3
Note (?3 lt 0 )
0
Xt
years of experience
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Testing Qualitative Effects

• 1. Test for differences in intercept.
• 2. Test for differences in slope.
• 3. Test for differences in both intercept
  and slope.

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men Dt 1 women Dt 0
??et
Yt
?
??
?
???
Xt
?
??
Dt
?
??
Dt
Xt
1
2
3
4
intercept
H0 ??????????vs???1 ???????????
Testing for discrimination in starting wage.
b??????
3
?
t
T
?
4
Est
.
Var
b?
3
H0 ??????????vs???1 ???????????
slope
b??????
Testing for discrimination in wage increases.
4
?
t
T
?
4
Est
.
Var
b?
4
13
Testing? Ho ????? ??????????H1
otherwise
?
SSE
?
SSE
?
?
2
R
U
?
F
SSE
?
?
T
?
4
?
? , T-4
U
T
?
2
?
SSE
?
?
y
?
b
?
b
X
?
b
D
?
b
D
X
t
t
U
t
?
t
t
1
?
?
t
?
1
intercept and slope
and
T
2
?
SSE
?
?
y
?
?
b
?
b
X
R
2
t
1
t
?
t
1
14
The University Effect on House Prices

• A real estate economist collects data on two
  similar neighborhoods, one bordering a large
  state university, and one that is a neighborhood
  about 3 miles from the university.
• Records 1000 observations
• Dependent Variable House prices are given in
• Independent Variables
• SQFT is the number of square feet of living area.
  
• AGE are the house age (years)
• UTOWN 1 for homes near the university, 0
  otherwise
• USQFT SQFT ? UTOWN
• POOL 1 if a pool is present, 0 otherwise
• FPLACE 1 is a fireplace is present, 0 otherwise
   

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• We anticipate that all the coefficients in this
  model will be positive except , which is an
  estimate of the effect of age (or
  depreciation) on house price.
• The model R-squared 0.869 and the overall-F
  statistic value is F 1104.213.

  Parameter
Standard T for H0 Variable
DF Estimate Error Parameter0
Prob gt T   INTERCEP 1 24500
6191.721 3.957
0.0001 UTOWN 1 27453
8422.582 3.259 0.0012
SQFT 1 76.122
2.452 31.048 0.0001
USQFT 1 12.994
3.321 3.913 0.0001
AGE 1 -190.086
51.205 -3.712 0.0002 POOL
1 4377.163 1196.692
3.658 0.0003
FPLACE 1 1649.176
971.957 1.697 0.0901
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Based on these regression estimates, what do we
conclude?

• We estimate the location premium, for lots near
  the university, to be 27,453
• We estimate the price per square foot to be
  89.11 ( 76.122 12.994) for houses near
  the university, and 76.12 for houses in other
  areas.
• We estimate that houses depreciate 190.09 per
  year
• We estimate that a pool increases the value of a
  home by 4377.16
• We estimate that a fireplace increases the value
  of a home by 1649.17

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Are Two Regressions Equal?
Chow Test (there are two alternative ways)
I. Restricted versus Unrestricted Models
men Dt 1 women Dt 0
yt ?1 ?2 Xt ?3 Dt ?4 Dt Xt et
H0 ?3 ?4 0 vs. H1 otherwise
yt wage rate
Xt years of experience
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II. Get SSEU separately
(running three regressions)
Forcing men and women to have same ?1, ?2.
Everyone
SSER
yt ?1 ?2 Xt et
Allowing men and women to be different.
Men only
SSEm
ytm ?1 ?2 Xtm etm
Women only
SSEw
ytw ?1 ?2 Xtw etw
(SSER ? SSEU)/J
J restrictions
F
SSEU /(T?K)
Kunrestricted coefs.
where SSEU SSEm SSEw
J 2 K 4
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Interaction Variables

• 1. Interaction Dummies
• 2. Polynomial Terms (special case of
  continuous interaction)
• 3. Interaction Among Continuous Variables

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Interactions Between Qualitative Factors

• Suppose we are estimating a wage equation, in
  which an individuals wages are explained as a
  function of their experience, skill, and other
  factors related to productivity.
• It is customary to include dummy variables for
  race and gender in such equations.
• Including just race and gender dummies will not
  capture interactions between these qualitative
  factors. Special wage treatment for being
  white and male is not captured by separate
  race and gender dummies.
• To allow for such a possibility consider the
  following specification, where for simplicity we
  use only experience (EXP) as a productivity
  measure

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Wage ?1 ?2 EXP ? 1 RACE ? 2 SEX ? (RACE
? SEX) e
where
? 1 measures the effect of race  ? 2
measures the effect of gender ? measures the
effect of being white and male.
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1. Interaction Dummies
Wage Gap between Men and Women
yt wage rate Xt experience
For men? Mt 1. For women? Mt 0.
For black? Bt 1. For nonblack? Bt 0.
No Interaction wage gap assumed the same
yt ?1 ?2 Xt ?3 Mt ?4 Bt et
Interaction wage gap depends on race
yt ?1 ?2 Xt ?3 Mt ?4 Bt ?5 Mt Bt et
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2. Polynomial Terms
Polynomial Regression
yt income Xt age
Linear in parameters but nonlinear in variables
yt ?1 ?2 X t ?3 X2t ?4 X3t et
yt
90
70
X t
50
60
80
20
30
40
People retire at different ages or not at all.
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Polynomial Regression
yt income Xt age
yt ?1 ?2 X t ?3 X2t ?4 X3t et
Rate income is changing as we age
Slope changes as X t changes.
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3. Continuous Interaction
Exam grade f(sleepZt , study timeBt)
yt ?1 ?2 Zt ?3 Bt ?4 Zt Bt et
Sleep and study time do not act independently.
More study time will be more effective when
combined with more sleep and less effective when
combined with less sleep.
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continuous interaction
Exam grade f(sleepZt , study timeBt)
yt ?1 ?2 Zt ?3 Bt ?4 Zt Bt et
Your studying is more effective with more sleep.
Your mind sorts things out while you sleep (when
you have things to sort out.)
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Exam grade f(sleepZt , study timeBt)
If Zt Bt 24 hours, then Bt (24 ? Zt)
yt ?1 ?2 Zt ?3 Bt ?4 Zt Bt et
yt ?1 ?2 Zt ?3(24 ? Zt) ?4 Zt (24 ? Zt) et
yt (?124 ?3) (?2??324 ?4)Zt ? ?4Z2t et
yt ?1 ?2 Zt ?3 Z2t et
Sleep needed to maximize your exam grade
where ?2 gt 0 and ?3 lt 0
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Qualitative Variables with Several Categories

• Many qualitative factors have more than two
  categories.
• Examples are region of the country (North, South,
  East, West) and level of educational attainment
  (less than high school, high school, college,
  postgraduate).
• For each category we create a separate binary
  dummy variable.
• To illustrate, let us again use a wage equation
  as an example, and focus only on experience and
  level of educational attainment (as a proxy for
  skill) as explanatory variables.

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Define dummies for educational attainment as
follows
Specify the wage equation as
Wage ?1 ?2 EXP ? 1 E1 ? 2 E2 ? 3 E3 e
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• First notice that we have not included all the
  dummy variables for educational attainment.
  Doing so would have created a model in which
  exact collinearity exists.
• Since the educational categories are exhaustive,
  the sum of the education dummies is equal to 1.
  Thus the intercept variable, is an exact
  linear combination of the education dummies.
• The usual solution to this problem is to omit one
  dummy variable, which defines a reference group,
  as we shall see by examining the regression
  function,

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• ?1 measures the expected wage differential
  between workers who have a high school diploma
  and those who do not.
• ?2 measures the expected wage differential
  between workers who have a college degree and
  those who did not graduate from high school, and
  so on.

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• The omitted dummy variable, E0, identifies those
  who did not graduate from high school. The
  coefficients of the dummy variables represent
  expected wage differentials relative to this
  group.
• The intercept parameter ?1 represents the base
  wage for a worker with no experience and no high
  school diploma.
• Mathematically it does NOT matter which dummy
  variable is omitted, although the choice of E0 is
  convenient in the example above.
• If we are estimating an equation using geographic
  dummy variables, N, S, E and W, identifying
  regions of the country, the choice of which dummy
  variable to omit is arbitrary.