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Chapter 9 Dummy (Binary) Variables 9.1 Introduction The multiple regression model
1
2
2
3
3
t^
t^
t^
K
tK
t
y
x
x
x
e
β + β
+…+ β
Assumption MR1 is
1
2
2
,^
t^
t^
K
tK
t
y
x
x
e
t^
T
= β + β
L
K
Assumption 1 defines the statistical model that we assume is appropriate for
all
T
of
the observations in our sample.
One part of the assertion is that the parameters of the
model,
β
, are the same for each and every observation. k
Slide 9.
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Recall that
β
k^
= the change in
E
( y
) when t
x
tk
is increased by one unit, and all other
variables are held constant^ =
(other variables held constant)
(^
)^
(^
t^
t
tk
tk
E y
E y
x
x
Assumption 1 implies that for each of the observations
t
T
the effect of a one
unit change in
x
tk
on
E
( y
) is exactly the same. t
If this assumption does not hold, and if the parameters are not the same for all theobservations, then the meaning of the least squares estimates of the parameters inequation 9.1.1 is not clear.
Undergraduate Econometrics, 2
nd Edition –Chapter 9
The Use of Intercept Dummy Variables
For the present, let us assume that the size of the house,
S
, is the only relevant variable
in determining house price,
P
. Specify the regression model as
1
2
t^
t^
t
P
S
e
β + β
In this model
is the value of an additional square foot of living area, and 2 β
1 β
is the
value of the land alone.
Dummy variables are used to account for qualitative factors in econometric models.They are often called
binary
or
dichotomous
variables as they take just two values,
usually 1 or 0, to indicate the presence or absence of a characteristic.
Undergraduate Econometrics, 2
nd Edition –Chapter 9
That is, a dummy variable
D
is
if property is in the desirable neighborhood
if property is not in the desirable neighborhood
t D
Adding this variable to the regression model, along with a new parameter
δ
, we obtain
1
2
t^
t^
t^
t
P
D
S
e
β + δ
The
regression function
is
1
2
1
2
(^
)^
when
(^
)^
when
t^
t
t
t^
t
S
D
E P
S
D
β
β
Adding the dummy variable
D
t^
to the regression model creates a
parallel shift
in the
relationship by the amount
δ
A dummy variable like
D
t^
that is incorporated into a regression model
to capture a
shift in the intercept as the result of some qualitative factor
is an
intercept dummy
variable
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Slide 9.
Undergraduate Econometrics, 2
nd Edition –Chapter 9
t^
t^
t^
t
t^
t
S
D
E P
S
S D
S
D
(^
)^
1
2
1
2
1
2 (^
)^
when
(^
)^
when
t^
t
β
= β + β
β
In the desirable neighborhood, the price per square foot of a home is (
β
2
γ
); it is
β
2
in
other locations.
We would anticipate that
γ
, the difference in price per square foot in the two locations,
is positive, if one neighborhood is more desirable than the other.
The effect of a change in house size on price is.
(^22)
when
(^
when
t
t
t
t
D
E P
D
S
β
β
A test of the hypothesis that the value of a square foot of living area is the same in thetwo locations is carried out by testing the null hypothesis
against the
alternative
. I
0
H
γ =
1
:^
H
γ ≠
Undergraduate Econometrics, 2
nd Edition –Chapter 9
In this case, we might test
0
H
γ =
against
1
H
γ >
, since we expect the effect to be
positive. •^
If we assume that house location affects
both
the intercept and the slope, then both
effects can be incorporated into a single model. The resulting regression model is
1
2
(^
t^
t^
t^
t^
t^
t
P
D
S
S D
e
β + δ
In this case the regression functions for the house prices in the two locations are
1
2
1
2
(^
)^
(^
)^
when
(^
)^
when
t^
t
t
t^
t
S
D
E P
S
D
β
β
House prices are given in $; size (SQFT) is the number of square feet of living area.
Also recorded are the house age (years)
UTOWN = 1 for homes near the university, 0 otherwise
POOL = 1 if a pool is present, 0 otherwise
FPLACE = 1 is a fireplace is present, 0 otherwise
The economist specifies the regression equation as
(^
1
1
2
3
2
3
t^
t^
t^
t^
t
t^
t^
t^
t
PRICE
UTOWN
SQFT
SQFT
UTOWN
AGE
POOL
FPLACE
e
= β + δ
×
β
Undergraduate Econometrics, 2
nd Edition –Chapter 9
We anticipate that all the coefficients in this model will be positive except
3 β
, which is
an estimate of the effect of age, or depreciation, on house price.
-^
Using 481 houses not near the university (UTOWN = 0) and 519 houses near theuniversity (UTOWN = 1). The estimated regression results are shown in Table 9.2.
-^
The model
and the overall-
F
statistic value is
2
R
F
Table 9.2 House Price Equation Estimates
Parameter
Standard
T for
H0:
Variable
DF
Estimate
Error
Parameter=
Prob
>
|T|
INTERCEP
1
24500
6191.
3.
0.
UTOWN
1
27453
8422.
3.
0.
SQFT
1
76.
2.
31.
0.
USQFT
1
12.
3.
3.
0.
AGE
1
-190.
51.
-3.
0.
POOL
1
4377.
1196.
3.
0.
FPLACE
1
1649.
971.
1.
0.
The estimated regression function for the houses near the university is
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Common Applications of Dummy Variables
In this section we review some standard ways in which dummy variables are used. Payclose attention to the interpretation of dummy variable coefficients in each example. 9.5.1 Interactions Between Qualitative Factors •
Suppose we are estimating a wage equation, in which an individual’s wages areexplained as a function of their experience, skill, and other factors related toproductivity.
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 thesequalitative 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 forsimplicity we use only experience (
EXP
) as a productivity measure,
(^
1
2
1
2
WAGE
EXP
RACE
SEX
RACE
SEX
e
= β + β
×
where
white
male
RACE
SEX
nonwhite
female
(^
(^
(^
1
1
2
2
1
1
2
1
2
2
1
2
(^
EXP
white
male
EXP
white
female
E WAGE
EXP
nonwhite
male
EXP
nonwhite
female
⎧ β + δ + δ + γ + β
β + δ
β + δ
⎪ ⎪β + β
measures the effect of race 1 δ
measures the effect of gender 2 δ
measures the effect of being “white” and “male.” γ
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Specify the wage equation as
1
2
1
1
2
2
3
3
WAGE
EXP
E
E
E
e
β + β
First notice that we have not included all the dummy variables for educationalattainment. Doing so would have created a model in which
exact collinearity
exists.
Since the educational categories are exhaustive, the sum of the education dummies
. Thus the “intercept variable”
, is an exact linear
combination of the education dummies.
0
1
2
3
E
E
E
E
1
x
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,
Undergraduate Econometrics, 2
nd Edition –Chapter 9
(^
(^
(^
1
3
2
1
2
2
1
1
2
1
2
p
ostgraduate degee college degree
(^
)^
high school diplomaless than high school
EXPEXP
E WAGE
EXP
EXP
⎧ β + δ
β + δ
β + δ
⎪ ⎪β + β⎩
δ
1
measures the expected wage differential between workers who have a high school diploma and those who do not.
δ
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.
The omitted dummy variable,
E
, identifies those who did not graduate from high 0
school. The coefficients of the dummy variables represent expected wage differentials relative to
this group.
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Undergraduate Econometrics, 2
nd Edition –Chapter 9
Explanatory variables would include the price of Royal Oak, the price of competitivebrands (Kingsford and the store brand), the prices of complementary goods (charcoallighter fluid, pork ribs and sausages) and advertising (newspaper ads and coupons).
We may also find strong seasonal effects.
Thus we may want to include either monthly dummies, (for example AUG=1 if monthis August, AUG=0 otherwise), or seasonal dummies (SUMMER=1 if month = June,July or August; SUMMER=0 otherwise) into the regression 9.5.3b Annual Dummies •
Annual dummies are used to capture year effects not otherwise measured in a model.
Real estate data are available continuously, every month, every year. Suppose we havedata on house prices for a certain community covering a 10-year period.
To capture macroeconomic price effects include annual dummies (D99=1 if year =1999; D99 = 0 otherwise) into the hedonic regression model
9.5.3c Regime Effects •
An economic regime is a set of structural economic conditions that exist for a certainperiod.
The investment tax credit was enacted in 1962 in an effort to stimulate additionalinvestment. The law was suspended in 1966, reinstated in 1970, and eliminated in theTax Reform Act of 1986.
Thus we might create a dummy variable
ITC
otherwise
A macroeconomic investment equation might be
I^
1
2
3
1
t^
t^
t^
t
Undergraduate Econometrics, 2
nd Edition –Chapter 9
t
NV
ITC
GNP
GNP
e
−
β + δ
