Dummyvariable1

dummy

1. Dummy Variable
Regression Models

2. What is Dummy Variable?
•Variablesthatareessentiallyqualitativein
nature(or)variablesthatarenotreadily
quantifiable
•Examples:gender,maritalstatus,race,
colour,religion,nationality,geographical
location,political/policychanges,party
affiliation

3. Other Names for Dummy Variable
Indicatorvariables
Binaryvariables
Categoricalvariables
Dichotomousvariables
Qualitativevariables

4. Why Dummy Variable Regression?
•To include qualitative variables as an
explanatory variable in the regression
model
•Example: If we want to see whether gender
discrimination has any influence on
earnings, apart from other factors

5. How to quantifyqualitative aspect?
•Byconstructingartificialvariablesthattake
onvaluesof1or0(zero)
•1indicatespresenceofthatattribute
•0indicatesabsenceofthatattribute
•Example:
(1)Gender=1iftherespondentisfemale
=0iftherespondentismale
(2)Time=1ifwartime;0ifpeacetime
•Herevariableswithvalues1and0are
calleddummyvariables

6. Types of Dummy Variable Models
(1)AnalysisofVariance(ANOVA)Model:All
explanatoryvariablesaredummyvariables
(2)AnalysisofCovariance(ANCOVA)Model:
Mixofquantitativeandqualitative
explanatoryvariables

7. ANOVA Model
•Supposewewanttomeasureimpactof
GENDERonwages/employeecompensation
•Inparticular,weareinterestedtoknow
whether femaleemployees are
discriminatedagainsttheirmale
counterparts
•Genderisnotstrictlyquantifiable

8. •Hence,wedescribegenderusingdummy
variable
D = 1 if male respondent
= 0 if female respondent [reference group]
Let the regression model as
Y
i= + D + u
i (1)
(where Y-Monthly salary)

9. •Thisspecificationhelpsustoseewhether
gendermakesdifferenceinsalary.
•Interpretationofmodel(1):
•Takingexpectationof(1)onbothsides,weget
•Meansalaryofmaleas
E(Y
i/D=1)=+
•Meansalaryoffemaleas
E(Y
i/D=0)=

10. •Note,meansalaryoffemaleisgivenby
intercept
•Coefficienttellsbyhowmuchmeansalary
ofmaleworkersdifferfrommeansalaryof
femaleworkers(or)simplydifferencein
averagesalarybetweenmen&women-
calleddifferentialinterceptcoefficient
•isattachedtocategorywhichisassigned
dummyvariablevalueof1(heremale)

11. •Intercept()belongstothecategoryfor
whichzerodummyvariablevalueisassigned
(herefemale)
•Thecategorywhichisassignedzerodummyis
knownasbenchmark/control/reference
category
•Interceptvaluerepresentsmeanvalueof
benchmarkcategory
•Allcomparisons(with)aremadeinrelation
tobenchmarkcategory

12. •Hypothesis testing:
•Done in the usual way
•H
0:=0[Nogenderdiscriminationinsalary
determination/nostatisticallysignificant
differenceinsalariesbetweenmalesand
females]
•H1:0[Genderdiscriminationispresentin
salarydetermination]
•Uset–statistics
•Ifissignificantlydifferentfromzero,wecan
acceptalternatehypothesis

13. Example: Cross Section Data on Monthly Wages and Gender
Y D Y
D Y D
1345 0 1566 0 2533 1
2435 1 1187 0 1602 0
1715 1 1345 0 1839 0
1461 1 1345 0 2218 1
1639 1 2167 1 1529 0
1345 0 1402 1 1461 1
1602 0 2115 1 3307 1
1144 0 2218 1 3833 1
1566 1 3575 1 1839 1
1496 1 1972 1 1461 0
1234 0 1234 0 1433 1
1345 0 1926 1 2115 0
1345 0 2165 0 1839 1
3389 1 2365 0 1288 1
1839 1 1345 0 1288 0
981 1 1839 0Male 26
1345 0 2613 1Female 23

14. Regression Results:
•Y = 1518.696 + 568.227 D
t:(12.394)(3.378) R
2
=0.195; F=11.410
Female Male
 (=1518.696)
+
=568.23
(=2086.923)
Y
Nos.

15. •Resultsshowmeansalaryoffemaleworkers
isaboutRs.1519
•Meansalaryofmaleworkersisincreasedby
Rs.568(i.e.1519+568=2087)
•tstatisticsrevealthatmeansalaryofmaleis
statisticallysignificantlyhigherbyabout
Rs.568

16. Does conclusion of model change if we
interchange dummy values?
Suppose Y
i= + D + u
i
where Y = Hourly wage
D= Gender (1= Male; 0 –Female)
Now,ifweinterchangedummyvaluesas(1=Female;0-
Male),itwillnotchangeoverallconclusionoforiginal
model(seefigure)
Onlychangeis,now“otherwise”categoryhasbecome
benchmarkcategoryandallcomparisonsaremadein
relationtothiscategory
•Hence, choice of benchmark category () is strictly up to
the researcher

17. Extension of ANOVA Model
•Canbeextendedtoincludemorethanone
qualitativevariable
Y
i= + 
1D
1+ 
2D
2+ u
i
where Y = Hourly wage
D
1= Marital status (1= married; 0 -otherwise)
D
2= Region of residence (1= south; 0 –otherwise)
Which is the benchmark category here?
Unmarried, non-south residence

18. •Mean hourly wages of benchmark category is 
•Mean wages of those who are married is + 
1
•Mean wages of those who live in south is + 
2

20. ANCOVA Model
•Consistsamixtureofqualitativeand
quantitativeexplanatoryvariables
•Suppose,inouroriginalmodel(1)weinclude
numberofyearsofexperienceasanadditional
variable
•Nowwecanraiseonemorequestion:between
2employeeswithsameexperience,istherea
genderdifferenceinwages?

21. •We can express regression model as
Y
i= 
1+ 
2D + X
i+ u
i
where D is dummy; X
iis experience variable
•Now, mean salary of male is
E (Y
i/D=1,X) = 
1+ 
2+ X
i
•Mean salary of female is
E (Y
i/D=0,X) = 
1+ X
i
•Slope is same for both categories (male &
female), only intercept differs
Slope

22. What does common slope mean?
•
2 measures average difference in salary between
male and female, given the same level of experience
•Ifwetakeafemaleandmalewithsamelevelsof
experience,
1+
2representssalaryofmale,on
average,and
1salaryoffemaleonaverage
•Notethatsincewecontrolledforexperienceinthe
regression,thewagedifferentialcan’tbeexplainedby
differentaveragelevelsofexperiencebetweenmale
andfemale

23. •Hence,wecanconcludethatwagedifferentialis
duetogenderfactor

24. Diagrammatic Explanation
X
Y

1+ X
i

1+
2+X
i
Constant Term 
1–intercept for base group;

1 + 
2–intercept for male; and

2measures the difference in intercept

1

2
1+2
Slope

25. Regression Estimation Results:
Y (Cap) = 1366.267 + 525.632 D +19.807 X
(8.534)(3.114)1.456)
R
2
=0.48; F = 6.901
Interceptforfemale(base)Group
1=1366.27.It
measuresmeansalaryoffemale
InterceptformaleGroup,
1+
2=1891.90.It
measuresmeansalaryofmale,ofwhich525.63(
2)is
averagedifferenceinsalarybetweenmaleandfemale
(i.e.19.81)–asno.ofyearsofexperiencegoesup
by1year,onaverage,aworkers(maleorfemale)
salarygoesupbyRs.19.81

26. 
2–differenceininterceptis525.63andis
statisticallysignificantat5%level.
Therefore,wecanrejectthenullhypothesis
ofnogenderdifferential

27. Example: Several qualitative variables, with
some having more than two category:
•Example:Consumptionfunctionanalysis.
•Supposetherearethreequalitativefactors:
gender,ageofhouseholdheadandeducation
levelofhead.
•Definedummyvariablesas:
D
1=1ifmaleand=0otherwise
D
2=1ifage<25and=0otherwise
D
3=1ifagebetween25and50and=0otherwise
D
4=1ifhighschooleducationand=0otherwise
D
5=1ifH.sc.,degreeandaboveand=0otherwise

28. Base or Reference Groups:
Regression Model:
C
t= + Y
t+ 
1D
1+ 
2D
2+ 
3D
3+ 
4D
4 + 
5D
5 + u
t
-intercept for female head of household
-intercept term if age of head is above 50 years
-intercept term if head’s education is below high school
In short represents female head of household aged above
50 years and with below high school education

29. Differential intercepts or mean
compensation for other groups:
+ 
1-for male household head
+ 
2 –for age is less than 25 years
+ 
3 -for age between 25 and 50 years
+ 
4 –for high school education
+ 
5 –for above high school education
If the household head is male with age 40 years and
high school education, what is the intercept?
+ 
1+ 
3+ 
4

30. Interactions Involving Dummy Variables
•Consider the following model:
Y
i= 
1+ 
2D
2i+ 
3D
3i+ X
i+ u
i -----(1)
Y
i= Hourly wage
X
i= Education (years of schooling)
D
2= 1 if female, 0 if male [GENDER]
D
3= 1 if black, 0 if white [RACE]
Notethatinthismodeldummyvariablesare
interactiveinnature.How?

31. •Here, if mean salary is higher for female than for
male, this is so whether they (female) are black or
white
•Similarly,ifmeansalaryislowerforblack,thisisso
whetherthey(black)aremaleorfemale
•Implication:EffectofD
2andD
3onYmaynotbe
simplyadditiveasin(1)butmultiplicativeasbelow
Male Female Black White
BlackWhiteBlackWhiteMaleFemaleMaleFemale
Gender
Race

32. Y
i= 
1+ 
2D
2i+ 
3D
3i+ 
4 (D
2iD
3i)+ X
i+ u
i --(2)
Eq(2)includesexplicitlyinteractionbetweenGENDER&
RACE,i.e.D
2iD
3i

2–differential effect of being a female (gender alone)

3 –differential effect of being a black (race alone)

4–differential effect of being a black female (g & r)

1 –Male white (base category)
Note: While running (2), simply multiply D
2iD
3i values
Eq(2)isadifferentwayoffindingwagedifferentials
acrossallgender-racecombinations

33. Inotherwordsinteractivemodel(eq.2)allowsus
toobtainestimatedwagedifferentialamongall4
groups(male,female,black&white).How?
(i) Black Female (Y
i/D
2i=1, D
3i=1, X
i)

1+ 
2+ 
3 + 
4
(ii) Black Male (Y
i/D
2i=0, D
3i=1, X
i)

1+ 
3
(iii) White Male (Y
i/D
2i=0, D
3i=0, X
i)

1
(iv) White Female (Y
i/D
2i= 1, D
3i=0, X
i)

1+ 
2

34. Male Female Married (M)Unmarried (UM)
M UM M UMMaleFemaleMaleFemale
Gender
Marital status