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Chapter 9
Dummy Variables
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 1
Chapter Contents




9.1 Introduction
9.2 The Use of Intercept Dummy Variables
9.3 Slope Dummy Variables
9.4 An Example: The University Effect on House
Price
 9.5 Common Applications of Dummy Variables
 9.6 Testing the Existence of Qualitative Effects
 9.7 Testing the Equivalence of Two Regression
Using Dummy Variables
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 2
9.1
Introduction
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 3
9.1
Introduction
The multiple regression model is
yt  1  2 xt 3  3 xt 3 
  K xtK  et
We explore the variables ways, dummy variables
can be included in a model and the different
interpretations that they bring.
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 5
9.1
Introduction
Assumptions of the multiple regression model
MR1. yt  1  2 xt 2  3 xt 3 
MR2. E(yt )  1  2 xt 2 
  K xtK  et
  K xtK  E (et )  0
MR3. var(yt )  var(et )  2
MR4. cov(yt , ys )  cov(et , es )  0
MR5. The values of x tK are not random and are not exact
linear functions of the other explanary variables
MR6. yt ~ N (1  2 xt 2 
Undergraduated Econometrics
  K xtK ,  2 )  et ~ N (0,  2 )
Chapter 9: Dummy Variables
Page 6
9.2
The Use of Intercept Dummy
Variables
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 7
9.2
The Use of
Intercept Dummy
Variables
Dummy variables allow us to construct models in
which some or all regression model parameters,
including the intercept, change for some
observations in the sample
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 7
9.2
The Use of
Intercept Dummy
Variables
Consider a hedonic model to predict the value of a
house as a function of its characteristics:
– size
– Location
– number of bedrooms
– age
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 8
9.2
The Use of
Intercept Dummy
Variables
Consider the square footage at first:
PRICE  β1  β 2 St  e
– β2 is the value of an additional square foot of
living area and β1 is the value of the land alone
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 9
9.2
The Use of
Intercept Dummy
Variables
How do we account for location, which is a qualitative
variable?
– Dummy variables are used to account for qualitative
factors in econometric models
– They are often called binary or dichotomous variables,
because they take just two values, usually one or zero,
to indicate the presence or absence of a characteristic or
to indicate whether a condition is true or false
– They are also called dummy variables, to indicate that
we are creating a numeric variable for a qualitative,
non-numeric characteristic
– We use the terms indicator variable and dummy
variable interchangeably
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 10
9.2
The Use of
Intercept Dummy
Variables
Generally, we define an indicator variable D as:
1 if characteristic is present
D
0 if characteristic is not present
– So, to account for location, a qualitative
variable, we would have:
1 if property is in the desirable neighborhood
D
0 if property is not in the desirable neighborhood
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 11
9.2
The Use of
Intercept Dummy
Variables
Adding our indicator variable to our model:
Pt  β1   Dt  β 2 St  et
– If our model is correctly specified, then:
 β1     β 2 St when D  1
E  Pt   
when D  0
β1  β 2 St
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 12
9.2
The Use of
Intercept Dummy
Variables
Adding the dummy variable causes a parallel shift
in the relationship by the amount δ
– An indicator variable like D that is incorporated
into a regression model to capture a shift in the
intercept as the result of some qualitative factor
is called an intercept indicator variable, or an
intercept dummy variable
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 13
9.2
The Use of
Intercept Dummy
Variables
Undergraduated Econometrics
FIGURE 9.1 An intercept dummy variable
Chapter 9: Dummy Variables
Page 14
9.3
Slope Dummy Variables
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 15
9.3
Slope Dummy
Variables
Suppose we specify our model as:
PRICE  β1  β 2 St    St Dt   et
– The new variable (S×D) is the product of
house size and the indicator variable
• It is called an interaction variable, as it
captures the interaction effect of location and
size on house price
• Alternatively, it is called a slope-indicator
variable or a slope dummy variable,
because it allows for a change in the slope of
the relationship
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 16
9.3
Slope Dummy
Variables
Now we can write:
E  Pt   β1  β 2 St    St Dt 
β1   β 2    St when D  1

when D  0
β1  β 2 St
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 17
9.3
Slope Dummy
Variables
Undergraduated Econometrics
FIGURE 9.2 (a) A slope dummy variable
(b) Slope- and intercept dummy variables
Chapter 9: Dummy Variables
Page 18
9.3
Slope Dummy
Variables
The slope can be expressed as:
E  Pt 
St
Undergraduated Econometrics
β 2  γ when Dt  1

when Dt  0
β 2
Chapter 9: Dummy Variables
Page 19
9.3
Slope Dummy
Variables
Assume that house location affects both the
intercept and the slope, then both effects can be
incorporated into a single model:
PRICE  β1  δDt  β 2 St  γ  St  Dt   et
– The variable (SQFTD) is the product of house
size and the indicator variable, and is called an
interaction variable
• Alternatively, it is called a slope-indicator
variable or a slope dummy variable
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 20
9.3
Slope Dummy
Variables
Now we can see that:
 β1  δ    β 2  γ  St
E  Pt   
β1  β 2 St
Undergraduated Econometrics
Chapter 9: Dummy Variables
when Dt  1
when Dt  0
Page 21
9.4
An Example: The University Effect
on House Prices
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 22
9.4
Slope Dummy
Variables
Suppose an economist specifies a regression
equation for house prices as:
PRICE  β1  δ1UTOWN  β 2 SQFT  γ  SQFT UTOWN 
β3 AGE  δ2 POOL  δ3 FPLACE  e
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 23
9.4
Slope Dummy
Variables
Suppose an economist specifies a regression
equation for house prices as:
PRICE  β1  δ1UTOWN  β 2 SQFT  γ  SQFT UTOWN 
β3 AGE  δ2 POOL  δ3 FPLACE  e
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 24
9.4
Slope Dummy
Variables
Undergraduated Econometrics
Table 9.1 Representative Real Estate Data Values
Chapter 9: Dummy Variables
Page 25
9.4
Slope Dummy
Variables
Undergraduated Econometrics
Table 9.2 House Price Equation Estimates
Chapter 9: Dummy Variables
Page 26
9.4
Slope Dummy
Variables
The estimated regression equation is for a house
near the university is:
PRICE   24.5  27.453   7.6122  1.2994  SQFT 
0.1901AGE  4.3772 POOL  1.6492 FPLACE
 51.953  8.9116SQFT  0.1901AGE
4.3772 POOL  1.6492 FPLACE
– For a house in another area:
PRICE  24.5  7.6122SQFT  0.1901AGE 
4.3772 POOL  1.6492 FPLACE
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 27
9.4
Slope Dummy
Variables
We therefore estimate that:
– The location premium for lots near the
university is $27,453
– The change in expected price per additional
square foot is $89.12 for houses near the
university and $76.12 for houses in other areas
– Houses depreciate $190.10 per year
– A pool increases the value of a home by
$4,377.20
– A fireplace increases the value of a home by
$1,649.20
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 28
9.5
Common Application of Dummy
Variables
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 29
9.5
Common
Application of
Dummy Variables
9.5.1
Interactions
Between
Qualitative
Factors
Consider the wage equation:
WAGE  β1  β 2 EXP  δ1 RACE  δ 2 SEX
 γ  RACE SEX   e
– The expected value is:
 β1  δ1  δ 2  γ   β 2 EXP WHITE - MALE

WHITE - FEMALE
 β1  δ1   β 2 EXP
E WAGE   
NONWHITE - MALE
 β1  δ 2   β 2 EXP
β  β EXP
NONWHITE - FEMALE
 1 2
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 30
9.5
Common
Application of
Dummy Variables
9.5.2
Qualitative
Factors with
Several
Categories
Many qualitative factors have more than two
categories.
To illustrate:
1 less than high school
E0  
0 otherwise
1 colledge degree
E2  
0 otherwise
1 high school diploma
E1  
0 otherwise
1 postgraduate degree
E3  
0 otherwise
WAGE  β1  β 2 EDUC  δ1SOUTH  δ 2 MIDWEST  δ3WEST  e
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 31
9.5
Common
Application of
Dummy Variables
9.5.2
Qualitative
Factors with
Several
Categories
Omitting one dummy variable defines a reference
group so our equation is:
 β1  δ3   β 2 EXP

 β1  δ 2   β 2 EXP
E WAGE   
 β1  δ1   β 2 EXP
β  β EXP
 1 2
postgraduatedegree
colledgedegree
highs chool diploma
less than high school
– The omitted dummy variable identifies the
reference
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 32
9.5
Common
Application of
Dummy Variables
9.5.3
Controlling for
Time
9.5.3a
Seasonal
Dummies
Indicator variables are also used in regressions
using time-series data
We may want to include an effect for different
seasons of the year
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 33
9.5
Common
Application of
Dummy Variables
9.5.3a
Annual
Dummies
9.5.3c
Regime Effects
In the same spirit as seasonal dummies, annual
dummies are used to capture year effects not
otherwise measured in a model
An economic regime is a set of structural
economic conditions that exist for a certain period
– The idea is that economic relations may behave
one way during one regime, but may behave
differently during another
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 34
9.5
Common
Application of
Dummy Variables
9.5.3c
Regime Effects
An example of a regime effect: the investment tax
credit:
1 if t  1962 -1965, 1970 -1986
ITCt  
0 otherwise
– The model is then:
INVt  β1  δITCt  β2GNPt  β3GNPt 1  et
– If the tax credit was successful, then δ > 0
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 35
9.6
Testing the Existence of Qualitative
Effects
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 36
9.6
Testing the
Existence of
Qualitative Effects
If the regression model assumptions hold, and the
error e are normally distributed, or if the errors are
mot normal but the sample is large, then the
testing procedures outlined in Chapters 7.5, 8.1
and 8.2 may be used to test for the presence of
qualitative effects.
9.6.1
Testing For a
Single
Qualitative
Effect
Tests for the presence of a single qualitative effect
can be based on the t-distribution.
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 37
9.6
Testing the
Existence of
Qualitative Effects
9.6.2
Testing Jointly
For the
Presence of
Several
Qualitative
Effect
If a model has more than one dummy variable,
representing several qualitative characteristic, the
significance of each, apart from the others, can be
tested using the t-test outlined in the previous
section.
To test the joint significance of all the qualitative
factors
WAGE  β1  β 2 EXP  δ1 RACE  δ 2 SEX
 γ  RACE SEX   e
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 38
9.6
Testing the
Existence of
Qualitative Effects
9.6.2
Testing Jointly
For the
Presence of
Several
Qualitative
Effect
We do it by testing the joint null hypothesis
H 0 :δ1  0, δ 2  0, γ  0
against the alternative that at least one of the
indicated parameters is not zero.
Use the F-test procedure
( SSER  SSEU ) / J
F
SSEU / (T  K )
We reject the null hypothesis if F≥ Fc, where Fc is
the critical value. Illustrated in Figure 8.1, for the
level of significance α.
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 39
9.7
Testing the Equivalence of Two
Regressions Using Dummy Variables
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 40
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
Suppose we have:
Pt  β1  δDt  β 2 St  γ  St  Dt   et
and for two locations:
Eq. 9.7.2
( 1   )  (  2   ) St desirable neighbourhood
E  Pt   
other neighbourhood
β1  β 2 St
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 41
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.1
The Chow Test
The Chow test is an F-test for the equivalence of
two regressions
Are there differences between the hedonic
regressions for the two neighborhood or not?’’
• If there are no differences, then the data from
the two neighborhoods can be pooled into one
sample, with no allowance made for differing
slope or intercept
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 42
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.1
The Chow Test
From(9.7.2), by testing H 0 :δ1  0, γ  0 , we are
testing the equivalence of the two regressions
Pt  1   2 St  et
Pt  1   2 St  et
Eq. 9.7.3
If we reject either or both of these hypotheses, then
the equalities 1  1 and  2   2 are not true, in which
case pooling the data together would be equivalent to
imposing constraints, or restrictions, which are not
true on the parameters of (9.7.3).
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 43
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.1
The Chow Test
The Chow test is an F-test for the equivalence of
two regressions
Are there differences between the hedonic
regressions for the two neighborhood or not?’’
• If there are no differences, then the data from
the two neighborhoods can be pooled into one
sample, with no allowance made for differing
slope or intercept
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 44
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.1
The Chow Test
Remark:
– The usual F-test of a joint hypothesis relies on the
assumptions MR1–MR6 of the linear regression
model
– Of particular relevance for testing the equivalence
of two regressions is assumption MR3, that the
variance of the error term, var(ei ) = σ2, is the same
for all observations
– If we are considering possibly different slopes and
intercepts for parts of the data, it might also be true
that the error variances are different in the two
parts of the data
• In such a case, the usual F-test is not valid
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 45
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
Table 9.3 Time Series Data on Real INV, V and K
9.7.2
An Empirical
Example Of
The Chow Test
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 46
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.2
An Empirical
Example Of
The Chow Test
The variables for each firm, in 1947 dollars, are
INV  gross investment in plant and equipment
V  value of the firm
K = stock of capital
A simple investment function is
INVt  1   2Vt   3 K t  et
Include an intercept dummy variable and a complete
set of slope dummy variables
INVt  β1  δ1 Dt  β 2Vt  δ 2 ( DtVt )  3 K t  δ3  Dt K t   et
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 47
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.2
An Empirical
Example Of
The Chow Test
The estimated restricted model with t-statistics in
parameters

I N V  17.8720  0.0152V  0.1436 K
(2.544)
(2.452)
(7.719)
SSER  16563.00
Unrestricted

I N V  9.9563  9.4469 D  0.0266V  0.0263( DV )  0.1517 K  0.0593( DK )
(0.421)
(0.328) (2.265)
(0.767)
(7.837)
(-0.507)
SSE  14989.82
R
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 48
9.7
Testing the
Equivalence of
Two Regressions
Using Dummy
Variables
9.7.2
An Empirical
Example Of
The Chow Test
Constructing the F-statistic
( SSER  SSEU ) / J
F
SSEU / (T  K )
(16563.00  14989.82) / 3

 1.1894
14989.82 / (40  6)
Since F≥ Fc, we cannot reject the null hypothesis.
The advantage of this approach to the Chow test is
that it does not require the construction of the dummy
and interaction variables.
Undergraduated Econometrics
Chapter 9: Dummy Variables
Page 49