Economics 20 - Prof,Anderson 1
Multiple Regression Analysis
y = b0 + b1x1 + b2x2 +,,, bkxk + u
5,Dummy Variables
Economics 20 - Prof,Anderson 2
Dummy Variables
A dummy variable is a variable that takes
on the value 1 or 0
Examples,male (= 1 if are male,0
otherwise),south (= 1 if in the south,0
otherwise),etc,
Dummy variables are also called binary
variables,for obvious reasons
Economics 20 - Prof,Anderson 3
A Dummy Independent Variable
Consider a simple model with one
continuous variable (x) and one dummy (d)
y = b0 + d0d + b1x + u
This can be interpreted as an intercept shift
If d = 0,then y = b0 + b1x + u
If d = 1,then y = (b0 + d0) + b1x + u
The case of d = 0 is the base group
Economics 20 - Prof,Anderson 4
Example of d0 > 0
x
y
{ d0
} b0
y = (b0 + d0) + b1x
y = b0 + b1x
slope = b1
d = 0
d = 1
Economics 20 - Prof,Anderson 5
Dummies for Multiple Categories
We can use dummy variables to control for
something with multiple categories
Suppose everyone in your data is either a
HS dropout,HS grad only,or college grad
To compare HS and college grads to HS
dropouts,include 2 dummy variables
hsgrad = 1 if HS grad only,0 otherwise;
and colgrad = 1 if college grad,0 otherwise
Economics 20 - Prof,Anderson 6
Multiple Categories (cont)
Any categorical variable can be turned into
a set of dummy variables
Because the base group is represented by
the intercept,if there are n categories there
should be n – 1 dummy variables
If there are a lot of categories,it may make
sense to group some together
Example,top 10 ranking,11 – 25,etc,
Economics 20 - Prof,Anderson 7
Interactions Among Dummies
Interacting dummy variables is like subdividing
the group
Example,have dummies for male,as well as
hsgrad and colgrad
Add male*hsgrad and male*colgrad,for a total of
5 dummy variables –> 6 categories
Base group is female HS dropouts
hsgrad is for female HS grads,colgrad is for
female college grads
The interactions reflect male HS grads and male
college grads
Economics 20 - Prof,Anderson 8
More on Dummy Interactions
Formally,the model is y = b0 + d1male +
d2hsgrad + d3colgrad + d4male*hsgrad +
d5male*colgrad + b1x + u,then,for example,
If male = 0 and hsgrad = 0 and colgrad = 0
y = b0 + b1x + u
If male = 0 and hsgrad = 1 and colgrad = 0
y = b0 + d2hsgrad + b1x + u
If male = 1 and hsgrad = 0 and colgrad = 1
y = b0 + d1male + d3colgrad + d5male*colgrad +
b1x + u
Economics 20 - Prof,Anderson 9
Other Interactions with Dummies
Can also consider interacting a dummy
variable,d,with a continuous variable,x
y = b0 + d1d + b1x + d2d*x + u
If d = 0,then y = b0 + b1x + u
If d = 1,then y = (b0 + d1) + (b1+ d2) x + u
This is interpreted as a change in the slope
Economics 20 - Prof,Anderson 10
y
x
y = b0 +
b1x
y = (b0 + d0) + (b1 + d1) x
Example of d0 > 0 and d1 <
0
d = 1
d = 0
Economics 20 - Prof,Anderson 11
Testing for Differences Across
Groups
Testing whether a regression function is
different for one group versus another can
be thought of as simply testing for the joint
significance of the dummy and its
interactions with all other x variables
So,you can estimate the model with all the
interactions and without and form an F
statistic,but this could be unwieldy
Economics 20 - Prof,Anderson 12
The Chow Test
Turns out you can compute the proper F statistic
without running the unrestricted model with
interactions with all k continuous variables
If run the restricted model for group one and get
SSR1,then for group two and get SSR2
Run the restricted model for all to get SSR,then
? ?? ? ? ?? ?
1
12
21
21
?
??
?
?
??
?
k
kn
S S RS S R
S S RS S RS S R
F
Economics 20 - Prof,Anderson 13
The Chow Test (continued)
The Chow test is really just a simple F test
for exclusion restrictions,but we’ve
realized that SSRur = SSR1 + SSR2
Note,we have k + 1 restrictions (each of
the slope coefficients and the intercept)
Note the unrestricted model would estimate
2 different intercepts and 2 different slope
coefficients,so the df is n – 2k – 2
Economics 20 - Prof,Anderson 14
Linear Probability Model
P(y = 1|x) = E(y|x),when y is a binary
variable,so we can write our model as
P(y = 1|x) = b0 + b1x1 + … + bkxk
So,the interpretation of bj is the change in
the probability of success when xj changes
The predicted y is the predicted probability
of success
Potential problem that can be outside [0,1]
Economics 20 - Prof,Anderson 15
Linear Probability Model (cont)
Even without predictions outside of [0,1],
we may estimate effects that imply a change
in x changes the probability by more than
+1 or –1,so best to use changes near mean
This model will violate assumption of
homoskedasticity,so will affect inference
Despite drawbacks,it’s usually a good
place to start when y is binary
Economics 20 - Prof,Anderson 16
Caveats on Program Evaluation
A typical use of a dummy variable is when
we are looking for a program effect
For example,we may have individuals that
received job training,or welfare,etc
We need to remember that usually
individuals choose whether to participate in
a program,which may lead to a self-
selection problem
Economics 20 - Prof,Anderson 17
Self-selection Problems
If we can control for everything that is
correlated with both participation and the
outcome of interest then it’s not a problem
Often,though,there are unobservables that
are correlated with participation
In this case,the estimate of the program
effect is biased,and we don’t want to set
policy based on it!
Multiple Regression Analysis
y = b0 + b1x1 + b2x2 +,,, bkxk + u
5,Dummy Variables
Economics 20 - Prof,Anderson 2
Dummy Variables
A dummy variable is a variable that takes
on the value 1 or 0
Examples,male (= 1 if are male,0
otherwise),south (= 1 if in the south,0
otherwise),etc,
Dummy variables are also called binary
variables,for obvious reasons
Economics 20 - Prof,Anderson 3
A Dummy Independent Variable
Consider a simple model with one
continuous variable (x) and one dummy (d)
y = b0 + d0d + b1x + u
This can be interpreted as an intercept shift
If d = 0,then y = b0 + b1x + u
If d = 1,then y = (b0 + d0) + b1x + u
The case of d = 0 is the base group
Economics 20 - Prof,Anderson 4
Example of d0 > 0
x
y
{ d0
} b0
y = (b0 + d0) + b1x
y = b0 + b1x
slope = b1
d = 0
d = 1
Economics 20 - Prof,Anderson 5
Dummies for Multiple Categories
We can use dummy variables to control for
something with multiple categories
Suppose everyone in your data is either a
HS dropout,HS grad only,or college grad
To compare HS and college grads to HS
dropouts,include 2 dummy variables
hsgrad = 1 if HS grad only,0 otherwise;
and colgrad = 1 if college grad,0 otherwise
Economics 20 - Prof,Anderson 6
Multiple Categories (cont)
Any categorical variable can be turned into
a set of dummy variables
Because the base group is represented by
the intercept,if there are n categories there
should be n – 1 dummy variables
If there are a lot of categories,it may make
sense to group some together
Example,top 10 ranking,11 – 25,etc,
Economics 20 - Prof,Anderson 7
Interactions Among Dummies
Interacting dummy variables is like subdividing
the group
Example,have dummies for male,as well as
hsgrad and colgrad
Add male*hsgrad and male*colgrad,for a total of
5 dummy variables –> 6 categories
Base group is female HS dropouts
hsgrad is for female HS grads,colgrad is for
female college grads
The interactions reflect male HS grads and male
college grads
Economics 20 - Prof,Anderson 8
More on Dummy Interactions
Formally,the model is y = b0 + d1male +
d2hsgrad + d3colgrad + d4male*hsgrad +
d5male*colgrad + b1x + u,then,for example,
If male = 0 and hsgrad = 0 and colgrad = 0
y = b0 + b1x + u
If male = 0 and hsgrad = 1 and colgrad = 0
y = b0 + d2hsgrad + b1x + u
If male = 1 and hsgrad = 0 and colgrad = 1
y = b0 + d1male + d3colgrad + d5male*colgrad +
b1x + u
Economics 20 - Prof,Anderson 9
Other Interactions with Dummies
Can also consider interacting a dummy
variable,d,with a continuous variable,x
y = b0 + d1d + b1x + d2d*x + u
If d = 0,then y = b0 + b1x + u
If d = 1,then y = (b0 + d1) + (b1+ d2) x + u
This is interpreted as a change in the slope
Economics 20 - Prof,Anderson 10
y
x
y = b0 +
b1x
y = (b0 + d0) + (b1 + d1) x
Example of d0 > 0 and d1 <
0
d = 1
d = 0
Economics 20 - Prof,Anderson 11
Testing for Differences Across
Groups
Testing whether a regression function is
different for one group versus another can
be thought of as simply testing for the joint
significance of the dummy and its
interactions with all other x variables
So,you can estimate the model with all the
interactions and without and form an F
statistic,but this could be unwieldy
Economics 20 - Prof,Anderson 12
The Chow Test
Turns out you can compute the proper F statistic
without running the unrestricted model with
interactions with all k continuous variables
If run the restricted model for group one and get
SSR1,then for group two and get SSR2
Run the restricted model for all to get SSR,then
? ?? ? ? ?? ?
1
12
21
21
?
??
?
?
??
?
k
kn
S S RS S R
S S RS S RS S R
F
Economics 20 - Prof,Anderson 13
The Chow Test (continued)
The Chow test is really just a simple F test
for exclusion restrictions,but we’ve
realized that SSRur = SSR1 + SSR2
Note,we have k + 1 restrictions (each of
the slope coefficients and the intercept)
Note the unrestricted model would estimate
2 different intercepts and 2 different slope
coefficients,so the df is n – 2k – 2
Economics 20 - Prof,Anderson 14
Linear Probability Model
P(y = 1|x) = E(y|x),when y is a binary
variable,so we can write our model as
P(y = 1|x) = b0 + b1x1 + … + bkxk
So,the interpretation of bj is the change in
the probability of success when xj changes
The predicted y is the predicted probability
of success
Potential problem that can be outside [0,1]
Economics 20 - Prof,Anderson 15
Linear Probability Model (cont)
Even without predictions outside of [0,1],
we may estimate effects that imply a change
in x changes the probability by more than
+1 or –1,so best to use changes near mean
This model will violate assumption of
homoskedasticity,so will affect inference
Despite drawbacks,it’s usually a good
place to start when y is binary
Economics 20 - Prof,Anderson 16
Caveats on Program Evaluation
A typical use of a dummy variable is when
we are looking for a program effect
For example,we may have individuals that
received job training,or welfare,etc
We need to remember that usually
individuals choose whether to participate in
a program,which may lead to a self-
selection problem
Economics 20 - Prof,Anderson 17
Self-selection Problems
If we can control for everything that is
correlated with both participation and the
outcome of interest then it’s not a problem
Often,though,there are unobservables that
are correlated with participation
In this case,the estimate of the program
effect is biased,and we don’t want to set
policy based on it!
