Multiple Regression Analysis
y = β0 + β1x1 + β2x2 + . . . βkxk + u
5. Dummy Variables
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Dummy Variables
!   A dummy variable (or “indicator variable” is a
variable that takes on the value 1 or 0
Examples:
!   male (=1 if male, 0 otherwise)
!   south (=1 if in the south, 0 otherwise)
!   year70 (=1 if year=1970, 0 otherwise)
!   Dummy variables are also called binary variables,
for obvious reasons
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A Dummy Independent Variable
!   Dummy variables can be quite useful in regression
!
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analysis
Consider the simple model with one continuous
variable (x) and one dummy (d)
y = β0 + δ0d + β1x + u
We can interpret d as an intercept shift
If d = 0, then y = β0 + β1x + u
If d = 1, then y = (β0 + δ0) + β1x + u
Think of the case where d = 0 as the base group
(or reference group)
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Example of δ0 > 0
y
y=(β0 + δ0) + β1x
d= 1
δ0
{
d=0
} β0
slope = β1
for both!
y = β0 + β1x
x
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Dummy Trap
!   Suppose we include a dummy variable called
!
!
!
!
female (=1 if female, 0 otherwise)
We could not also include a dummy variable
called male (=1 if male, 0 otherwise)
This is because for all i, female=1-male; 
perfect colinearity
Intuitively it wouldn’t make sense to have both,
we only need 2 intercepts for the 2 groups
In general, with n groups use n-1 dummies
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Dummies With log(y)
!   Not surprisingly, when y is in log form the
coefficients have a percentage interpretation
!   Suppose the model is:
log(y) = β0 + δ0d + β1x + u
!   100(δ0) = is (very roughly) the percentage change
in y between two groups holding x fixed
Example:
!   Regression of log of housing prices on dummy for
“heritage” and square feet
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Interpreting coefficient on dummy with
logs
Actually, it’s not correct to say that 100*(δ0) is the
percent change in y associated with a 1 unit
change in x. Because the model is non-linear, we
have to think of the RHS parameters as partial
derivatives of the LHS w.r.t. the RHS variables of
interest.
But when we take a derivative, we are asking what
the change in the LHS is as we make a small
change in one of the RHS variables, holding the
others constant.
Changing a dummy variable from 0 to 1is not a
small change. It’s a discrete jump.
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Interpreting coefficient on dummy with
log y
Changing d from 0 to 1 leads to a
100 * (eδ 0 − 1)% change in y.
Changing d from 1 to 0 leads to a
100 * (e −δ 0 − 1)% change in y.
For instance, if δ0=0.2, this implies that changing d
from 0 to 1 leads to a 22% increase in y.
Changing d from 1 to 0 leads to an 18% decrease
in y.
For small values of δ0, the “100*δ0%”
approximation will be close. Not so for larger
values.
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Interpreting coefficient on dummy with
log y
Things get slightly more complicated when we
obtain an estimate of δ0. You can’t just plug the
OLS estimate of δ0 into the formula on the
previous slide, as this will produce a biased
estimate of the effect of changing d from 0 to 1.
Instead, for an estimate of the effect of changing d
from 0 to 1 use the formula:
⎛ (δˆ0 − 12 (se(δˆ0 ))2 ) ⎞
100 * ⎜ e
− 1⎟ %
⎝
⎠
For more on quirks involving dummy variables, see
http://davegiles.blogspot.ca/2011/03/dummies-fordummies.html
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Dummies for Multiple Categories
1. Several dummy variables can be included in the
same equation, e.g.,
y = β0 + δ0d0 + δ1d1 + β1x + u
!   Here the base case is whatever is implied by d0 =
d1 = 0 (i.e. when all the dummy variables are zero)
Example:
log(hprice)= β0 + δ0heritage + δ1pool + β1sqft + u
!   The base case is a non-heritage house with no pool
!   δ0 gives information about the effect of heritage
status (vs. not) on house price; δ1 gives us
information about how a pool changes house value
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Dummies for Multiple Categories
2. We can use dummy variables to control for one
variable with more than two categories
!   Suppose everyone in your data is either a HS
dropout, HS grad only, or college grad (at least)
!   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
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Multiple Categories (cont)
!   Any categorical variable can be turned into a set
of dummy variables
!   However, because the base group is represented
by the intercept, if there are n categories there
should be n-1 dummy variables
Example: Could turn # of bedrooms into dummies
!   If there are a lot of categories, it may make sense
to group some of them together
Example: Age, usually divide into age categories
!   In fact, can convert continuous variables as well
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Interactions Among Dummies
!   Interacting dummies is like subdividing the group
Example: have dummies for male, as well as hsgrad
and colgrad
!   Could include the new variables male*hsgrad and
male*colgrad
!   We now have a total of 5 dummy variables or 6
categories
Base group is female HS dropouts
hsgrad female hsgrad
male*hsgrad male hsgrad
colgrad  female college grad male*colgrad  male college
grad
male  male hs dropouts
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More on Dummy Interactions
Formally, the model is y = β0 + δ1male + δ2hsgrad +
δ3colgrad + δ4male*hsgrad + δ5male*colgrad + β1x + u,
then, for example:
1. If male = 0 and hsgrad = 0 and colgrad = 0
!   y = β0 + β1x + u
2. If male = 0 and hsgrad = 1 and colgrad = 0
!   y =β0+δ2hsgrad+β1x+u=(β0+δ2) + β1x + u
3. If male = 1 and hsgrad = 0 and colgrad = 1
!   y =β0+δ1male+δ3colgrad+δ5male*colgrad+β1x +u
=(β0+δ1+δ3+δ5)+β1x +u
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Other Interactions with Dummies
!   Can also consider interacting a dummy variable, d,
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!
!
with a continuous variable, x
y = β0 + δ0d + β1x + δ1d*x + u
If d = 0, then y = β0 + β1x + u
If d = 1, then y = (β0 + δ0) + (β1 + δ1)x + u
i.e. the dummy variable allows both the intercept
and the slope to change depending on whether it is
“switched” on or off
Example: suppose δ0>0 and δ1<0
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Example of δ0 > 0 and δ1 < 0
y
y = β0 + β1x
d=0
d=1
y = (β0 + δ0) + (β1 + δ1) x
x
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Testing for Differences Across
Groups
!   Sometimes we will want to test whether a
regression function (intercept and slopes) differs
across groups
Example: Wage Equation
log(wage)=β0 + β1educ + β2exper + u
!   We might think that the intercept and slopes will
differ for men and women (e.g. discrimination), so
log(wage)=β0 + δ0male + β1educ + δ1male*educ +
β2exper + δ2male*exper + u
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Testing Across Groups (cont)
!   This can be thought of as simply testing for the
joint significance of the dummy and its
interactions with all other x variables
!   So, we could estimate the model with all the
interactions (unrestricted) and without (restricted)
and form the F statistic
!   However, forming all of the interaction variables
etc. can be unwieldy
!   There is an alternative way to form the F statistic
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The Chow Test
!   It turns out that we can compute the proper F
statistic without estimating the unrestricted model
!   In the general model with an intercept and k
continuous variables we can test for differences
across two groups as follows:
1. Run the restricted model for group one and get
SSR1, then for group two and get SSR2
2. Run the restricted model for all to get SSR, then
[
SSR − (SSR1 + SSR2 )] [n − 2(k + 1)]
F=
•
SSR1 + SSR2
k +1
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The Chow Test (continued)
!   Notice that the Chow test is really just a simple F
test for exclusion restrictions, but we’ve realized
that SSRur = SSR1 + SSR2
!   Unrestricted model is exactly the same as running
the restricted model for the groups separately
!   Note, we have k + 1 restrictions (each of the slope
coefficients and the intercept)
!   Note also, the unrestricted model would estimate 2
different intercepts and 2 different slope
coefficients, so the denominator df is n – 2k – 2
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Linear Probability Model
!   Binary variables can also be introduced on the
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LHS of an equation
e.g. y=1 if employed, 0 otherwise
In this setting the βj’s have a different
interpretation
As long as E(u|x)=0, then
E(y|x) = β0 + β1x1 + … + βkxk , or
P(y = 1|x) = β0 + β1x1 + … + βkxk
The probability of “success” is a linear function of
the x’s
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Linear Probability Model (cont)
!   So, the interpretation of βj is the change in the
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!
!
!
probability of success (y=1) when xj changes,
holding the other x’s fixed
If xj is education and y is as above, βj is the change
in the probability of being employed when
education increases by 1 year
Predicted y is the predicted probability of success
One potential problem is that our predictions can
be outside [0,1]
How can a probability be greater than 1?
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Linear Probability Model (cont)
!   Even without predictions outside of [0,1], we may
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estimate marginal effects that are greater than 1
and less than -1
Usually best to use values of independent
variables near the sample means
This model will violate assumption of
homoskedasticity
Problem?
Standard errors are not valid (correctable)
Despite drawbacks, it’s usually a good place to
start when y is binary
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Nonlinear Alternatives to LPM
!   If you go on to practice econometrics with limited
dependent variables (such as dummies), you will
learn about some non-linear models that may be
more appropriate
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e.g. logit, probit models
These are little more complicated to work with than
LPM (marginal effects differ from coefficient
estimates, as in other nonlinear models)
Advantages: They restrict predicted values to lie
between 0 and 1; don’t have built-in heteroskedasticity
like LPM
EViews and other packages can estimate such models
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Policy Analysis Using Dummy Variables
!   Policy analysts often use dummies to measure the
effect of a government program


e.g., What’s the effect of raising the minimum wage?
A simple before/after comparison could be done like
this
  Take a sample of low-wage firms (say, fast food restaurants) in
a state or province that is changing its minimum wage law.
  Sample restaurants before minimum wage increase, then again
after minimum wage increase (could use same restaurants or
different ones, as long as sampling is random)
  Estimate model of staffing levels, where Post is dummy
indicating whether the observation is for a firm in the postincrease regime or the pre-increase regime (=1 if post)
# employeesit = β 0 + β1Post it + uit
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Policy Analysis Using Dummy Variables
!   If zero conditional mean assumption holds, then
βˆ1 gives us the effect of increasing the minimum
wage on employment levels at these firms. What
sign do you expect?


Do you see any reason the zero conditional mean assumption
might fail?
What if something else changes as the policy regime changes?


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
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Economy gets better or worse.
Taxes on business change
Immigration increases size of local labour force
Any of these changes, if not controlled for, would be captured in u and would
be correlated with Post. violation of zero conditional mean assumption
Is there a way (other than controlling for all possible things that
might change simultaneously with minimum wage law) we can
deal with this problem?
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Policy Analysis Using Dummy Variables
!   It would be nice if we had a “control group”—a
bunch of similar firms existing under similar
circumstances that didn’t experiment the
minimum wage increase.

How about fast food restaurants on the other side of the
state/provincial border?
  Similar economic circumstances
  Similar labour supply issues
# employeesit = β 0 + β1Post it + β 2 NJ + β 3 Post * NJ it + uit
!   NJ=1 if firm is in NJ; =0 if firm is in Pennsylvania
!   Post*NJ is an interaction of the two dummies
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Policy Analysis Using Dummy Variables
!   What this analysis does is look at the effect of an
increase in the minimum wage in NJ, using fastfood restaurants in Pennsylvania as a control
group.
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We have to assume that other determinants of
employment that change at the same time as the
minimum wage change are occurring in both states and
with equal magnitudes (recessions, booms, etc.)
One way to reassure ourselves (other than looking at
data on the local economies) is to look at employment
trends in FF restaurants in both states. If they tend to
move together closely, that will provide some
reassurance about our “identifying assumption”
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Policy Analysis Using Dummy Variables
!   Note that the ideal experiment would let us observe NJ at
one point in time, simultaneously in 2 states of the world

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The state with no minimum wage increase
The state with minimum wage increase
!   Unfortunately we can’t observe these parallel universes;
with this approach we’re trying to do the next best thing
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We’re inferring a counterfactual (parallel universe where NJ
experiences no minimum wage increase), by observing what
happens in PA
The counterfactual for NJ assumes NJ follows the same trend as in
PA; then we just compare the actual (with min wage increase)
outcome in NJ to the (inferred) counterfactual for NJ
This gives us our estimate of the effect of increasing the minimum
wage.
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Policy Analysis Using Dummy Variables
!   Focus on this picture and see if you can deduce for
yourself what the effect of a minimum wage
increase in NJ is (assuming common trends in
employment in PA and NJ).
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Policy Analysis Using Dummy Variables
# employeesit = β 0 + β1Post it + β 2 NJ + β 3 Post * NJ it + uit
βˆ1, βˆ 2 < 0
βˆ1
βˆ 2
βˆ 0
βˆ 0 , βˆ 3 > 0
βˆ 3
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Policy Analysis Using Dummy Variables
# employeesit = β 0 + β1Post it + β 2 NJ + β 3 Post * NJ it + uit
!   Given this model, average employment at a firm is
β 0 in PA before min wage increase
  β 0 + β1 in PA after min wage increase
  β + β in NJ before min wage increase
0
2
  β 0 + β1 + β 2 + β 3 in NJ after min wage increase
!   Change in employment in NJ: β1 + β 3
!   Change in employment in PA: β1
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Note: This is the counterfactual change in employment for NJ
!   Diff-in-Diff (ΔNJ-ΔPA): β 3
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Policy Analysis Using Dummy Variables
!   This is called a “difference in difference” analysis.
βˆ 3 gives the effect of the minimum wage.
!   Card and Krueger (1994) studied a minimum wage law
change in NJ in 1992, using fast food restaurants in NJ and
eastern PA as the sample
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Famously found no effect of minimum wage on employment
Some follow-up studies have disputed their findings, though not
much evidence of a large effect of the minimum wage on
employment
One hypothesis: minimum wage was so low to begin with that it
was close to the equilibrium (market-clearing) wage.
Remains a point of empirical study and argument.
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Policy Analysis Using Dummy Variables
!   We can denote a generic diff-in-diff specification as
Outcomeit = β 0 + β1Post it + β 2Treat + β 3 Post _ Treat it + uit
!   It’s not hard to think of dozens of cases in policy analysis
where this type of approach could be used

Effect of changes in welfare generosity on
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


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employment
fertility
marriage
school performance of kids of recipients
Effect of change in divorce laws on
  divorce
  child outcomes
  domestic violence incidents
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Policy Analysis Using Dummy Variables
!   Note the crucial assumption: Common trends
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If employment in NJ and PA always move together; or (more
generally) if outcome variable in Treat=1 vs. Treat=0 always move
together, then this is likely a valid assumption
If employment and NJ and PA often move together but sometimes
randomly deviate (for reasons other than changes in minimum
wage laws) then assumption is more suspect
Does the minimum wage really increase employment?
  Seems unlikely theoretically
  A more sensible conclusion would be either that minimum wage has no effect
or small effect on employment; OR something else changed in NJ (or PA) at
the same time as minimum wage law change is driving their finding (i.e., their
results are subject to bias
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Caveats on Policy Analysis When Those
Studied Choose Whether to be Treated
!   Consider a job training program that unemployed
workers have the option of signing up for
!   If individuals choose whether to participate in a
program, this may lead to a self-selection problem
!   e.g. Those that apply for job training may be more
diligent or may have had to qualify
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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!
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