Independent Dummy Variables

8/17/2019 Independent Dummy Variables

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Regression when some of theRegression when some of the

regressors are qualitativeregressors are qualitative

Salary =f( education, experience,Salary =f( education, experience, sex, racesex, race ))

+E+E ain!charges of "ehicle=f( #ge,ain!charges of "ehicle=f( #ge, qualityquality) +E) +E


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$uantitative varia%les in$uantitative varia%les in

regressionregression

(&ummy "aria%les)(&ummy "aria%les) 'n regression analysis response varia%le is not only inuenced %y'n regression analysis response varia%le is not only inuenced %y

the quantitative varia%les %ut also %y many varia%les of interestthe quantitative varia%les %ut also %y many varia%les of interestthat are not quantitative %ut are qualitative such as gender, race,that are not quantitative %ut are qualitative such as gender, race,colourcolour

or example* holding all other factors constant, female collegeor example* holding all other factors constant, female college

professors are found to earn less than their male counterpartsprofessors are found to earn less than their male counterparts on whites are found to earn less than whiteson whites are found to earn less than whites his pattern may result from sex or racial discrimination his pattern may result from sex or racial discrimination $ualitative varia%les such as sex and race does inuence the$ualitative varia%les such as sex and race does inuence the

dependent varia%le and clearly should %e included among thedependent varia%le and clearly should %e included among the

explanatory varia%les!explanatory varia%les! $ualitative varia%les usually indicate the presence or a%sence of$ualitative varia%les usually indicate the presence or a%sence of

a quality such as male or female %lac- or white, literate ora quality such as male or female %lac- or white, literate orilliterate, ur%an or rural, etc the pro%lem is how to incorporateilliterate, ur%an or rural, etc the pro%lem is how to incorporatesuch varia%les in regression along with other quantitativesuch varia%les in regression along with other quantitativevaria%lesvaria%les


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$ualitative varia%les in regression$ualitative varia%les in regression

(&ummy "aria%les)(&ummy "aria%les)

.ne method of quantifying such attri%utes is %y constructing.ne method of quantifying such attri%utes is %y constructingarti/cial varia%les which ta-e on values of 0 or 1, 1 indicatingarti/cial varia%les which ta-e on values of 0 or 1, 1 indicatingthe a%sence of an attri%ute and 0 indicating the presence ofthe a%sence of an attri%ute and 0 indicating the presence ofthat attri%utethat attri%ute

"aria%les which assume such 1 and 0 values are called dummy"aria%les which assume such 1 and 0 values are called dummyvaria%les other names for such varia%les are indicatorvaria%les other names for such varia%les are indicatorvaria%les, %inary varia%les qualitative varia%les andvaria%les, %inary varia%les qualitative varia%les andcategorical varia%les!categorical varia%les!

2e can include dummy varia%les in a regression and conduct2e can include dummy varia%les in a regression and conduct

hypothesis tests, 3ust as we can any other quantitativehypothesis tests, 3ust as we can any other quantitativevaria%les! 4ut,varia%les! 4ut, interpretation of the coecient on the dummyinterpretation of the coecient on the dummyvariablevariable is somewhat di5erent than what we6ve seen %eforeis somewhat di5erent than what we6ve seen %efore


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7onsider a data on annual salary of male7onsider a data on annual salary of male

and female college teachers and yearsand female college teachers and years

of teaching experience!of teaching experience!

&e/ne two dummy varia%les &8 9&e/ne two dummy varia%les &8 9

&:,one for each category and the model&:,one for each category and the model

isis

; ;ii= annual salary of the ith= annual salary of the ith

college professorcollege professor


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&e/ne one dummy varia%les &8&e/ne one dummy varia%les &8

for male category and the modelfor male category and the model

isis

;i= annual salary of the ith ;i= annual salary of the ith

college professor college professor


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'mportant notes'mportant notes

The assin'ent of " and 1 -alue are ar.itrary

/ateory that is assined the -alue of " is often referred to as

control roup0 .ase cateory0 or o'itted cateory in the sense

that co'parisons are 'ade ith that cateory n this exa'ple

fe'ale roup

ntercept ter' α1 in the 'odel is introduced for the control

roup


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'nterpretation of results'nterpretation of results

ean salary for emale professorean salary for emale professor

ean salary for male professorean salary for male professor

( ) iiii X D X Y β α +==Ε

12 "0( ) ( )

iiii X D X Y β α α ++==Ε 212 10

he estimated equation %y using .S is ;= 8>!> + ?!?> &8+

>!@0 < (0?!8A)(?!B) (8!C>)

0) he estimated mean salary of female college teacher is=8>!>+ >!@0<

8) he estimated mean salary of male college teacher is=:0!1>+ >!@0<

:) he estimated mean salary of female college teacher with ?years of experience is = 8>!>+>!@0(?) =@0!>?


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est of hypothesis est of hypothesis

2α

21 α+α

7ompare the annual mean salary of

male and female college teachers4o5 α26"

( )

( )ns

2

22 7!"7!

"!$!

8-ar

8t =

−=

α

α−α=

ie sex has no effect on 'ean

salary of collee teachers

1α

emale

ale


9/32


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1010

'ntroduce dummy varia%le for'ntroduce dummy varia%le for

GE&ERGE&ER

2here2here

;i= 4ase salary of the ith salesperson ;i= 4ase salary of the ith salesperson


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1111

Regressions for dummyRegressions for dummy

varia%lesvaria%les

Regression for saleswomen&8=1

Regression forsalesmen&8=0


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1212

2α

21 α+α

1α

Saleswomen

Salesmen

2α

21 α+α 1α

Saleswomen

Salesmen

1α 1

α

Saleswomen

Salesmen


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1313

SCATTER PLOT

Salary vs experiene

E!PER"E#

403020100

S A L A R $

14

13

12

11

10

9

8

7

6

%E#&ER

'e(ale

(ale

# l i f h d l# l i f th d l


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#nalysis of the model#nalysis of the model

#s the regression coeFcient for gender is signi/cant(pvalue H !1@), so we conclude that the /rm doesdiscriminate against its saleswomen! #s coeFcient fordummy varia%le is positive so salesmen6s salary(present category) is more than salesmen6s salary

4o5 α26" ie 9ender has no influence on salary


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Estimation in regressionEstimation in regression

Estimated mean salary of Salesman with @ months ofexperience

Estimated mean salary of Saleswoman with @ months of experience

:!#7#

or #7#!)#(22")"(;7"$!"#8

=

=++=iY


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Regression 2ith wo $ualitative "aria%lesRegression 2ith wo $ualitative "aria%les7onsider the regression of the advertising expenditure7onsider the regression of the advertising expenditure

on the sales, type of /rm (on the sales, type of /rm (incorporated, notincorporated, not

incorporatedincorporated) pu%lic relation department and quality of) pu%lic relation department and quality ofsales management (sales management (high, lowhigh, low))

UDαDααY 33221 +++=

&2)1 in*rp*ra+e,

)0 O+-er.ise

&3)1 i' /ali+y *' sales (anae(en+ -i-

)0 O+-er.ise

#*+ in*rp*ra+e, 'ir( .i+- l*. /ali+y

*' sales (anae(en+ is COPAR"SO#COPAR"SO#

a+e*ry


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&ata analysis in ''#4&ata analysis in ''#4

http://c/Documents%20and%20Settings/ARIF/Desktop/example1.MTWhttp://c/Documents%20and%20Settings/ARIF/Desktop/example1.MTWhttp://c/Documents%20and%20Settings/ARIF/Desktop/example1.MTW


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Es+i(a+e, Reressi*n e/a+i*n is

$)4400016000&25500&3 "#TERPRETAT"O# Averae expen,i+re '*r ase a+e*ry

n*+ in*p*ra+e, .i+- l*. (anae(en+

is 44000

T-e expen,i+re is ,erease y 16000 i' +-e 'ir( is in*rp*ra+e,Expen,i+re is inrease y 5500

i' /ali+y *' sale (anae(en+ is -i-:

#*nsini'ian+ vale +ra+i* '*r &2 in,ia+es +-a+ +-ere is n* ,i''erene e+.een +-e

A,ver+isin expen,i+re *' in*rp*ra+e, an, n*+ in*rp*ra+e, 'ir(s

#*nsini'ian+ vale +ra+i* '*r &3 in,ia+es +-a+ +-ere is n* ,i''erene e+.een +-e

A,ver+isin expen,i+re '*r +-e 'ir(s -avin -i- *r l*. /ali+y sales (anae(en+

&8=0 incorporated&8=0 incorporated

=1 .therwise(i!e not incorporated) =1 .therwise(i!e not incorporated)

(&:=0 if quality of sales management high(&:=0 if quality of sales management high

=1 .therwise( ow sales management) =1 .therwise( ow sales management)


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Regression 2ith more than two categoriesSuppose that %ased on the cross sectional data we want toregress the annual expenditure on healthcare %y anindividual on the income and education of the individual!

Since the varia%le education is qualitative in nature weconsider three mutually exclusive levels of education (i)ess than high school (ii) Iigh School (iii) 7ollege

UDαDααY 33221 +++=

;i= #nnual expenditure on health care


20/32


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2121

'nterpretation of results'nterpretation of results

ean healthcare expenditure %y an individual having less than high school e

X D DY E β α +=== 132 )"0"


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TET !" H#P!THE$TET !" H#P!THE$0) 7omparison of mean expenditure on health care %y an0) 7omparison of mean expenditure on health care %y an

individual having high school education and less than highindividual having high school education and less than high

school education ( 4ase group)school education ( 4ase group)

8) 7omparison of mean expenditure on health care %y an8) 7omparison of mean expenditure on health care %y anindividual having college education and less than high schoolindividual having college education and less than high school

education ( 4ase group)education ( 4ase group)

"α542"

=

"α54 3" =

Expendie= -4.41 + 7.92D2 +10.5 D3 +0.361X

SE (3.195) (3.328) (7.813) (0.122)

t-ratio 2.38* 1.34 (2.95)

P-Value 0.037 0.206 0.013

TET !" H#P!THE$TET !" H#P!THE$


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TET !" H#P!THE$TET !" H#P!THE$ :) 7omparison of mean expenditure on health care %y an individual having high school:) 7omparison of mean expenditure on health care %y an individual having high school

education and collegeeducation and college

32" αα54 =

Unrestricted Model *estricted %odel

i3i32i21 UDαDααY ++++= X β 32" αα54 = 32

UDααY

=

2

i2i=

21

D D DWhere

X

+=

+++= β

Unrestricted ANOVA Restricted ANOVA

SOV DF SS

Regression

Error

Total

3

11

14

3!"#$

%1#1

31"4#

SOV DF SS

Regression

Error

Total

"

1"

14

3!1#"

%"#&

31"4#

( ) ( ) ( ) ( ) ns

URUR

UR RUR R

Edf ESS

Edf Edf ESS ESS F 2!1#"

#!

"1

11


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&K; "#R'#4E .R 7.L#R'G&K; "#R'#4E .R 7.L#R'G

2. REGRESS'. 'ES 2. REGRESS'. 'ES

2hen we use a regression model involving time series data, it2hen we use a regression model involving time series data, it

may happen that there is amay happen that there is a structural changestructural change in thein the

relationship %etween the regressand and the regressorsrelationship %etween the regressand and the regressors

4y structural change we mean that the value of the parameters4y structural change we mean that the value of the parametersof the model do not remain the same through the entire timeof the model do not remain the same through the entire time

periodperiod!! &ummy varia%le can %e used to /nd structural&ummy varia%le can %e used to /nd structural

changes in the regression parameters!changes in the regression parameters!

7oncurrent Larallel &issimilar Equal or

7oincident

E%uality of two regressionE

%uality of two regression


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E%uality of two regressionE%uality of two regression

lines by dummy variablelines by dummy variable

approach:approach:E


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2626

( ) ( )β βα α UD X D X2i i 2i i1 2 i1 2ii y = + + + + − − − −

;i = saving of a family


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he equality of two saving function can %e he equality of two saving function can %e

tested %y considering the hypothesistested %y considering the hypothesis


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S."S." && SSSS

RegRegErrorError

::0?0?

0?C>?C:>0?C>?C:>?:8@:8???:8@:8??

otal otal 0B0B A1111111A1111111

S."S." && SSSS

RegReg

ErrorError

00

0A0A

0@A1?@?00@A1?@?0

?>0B:>:B?>0B:>:B

otal otal 0B0B A1111111A1111111

Unrestricted %odel

( )β βα α UD X D X2i i 2i i1 2 i1 2i

y = + + + +*estricted%odel

iβα Xi1 1i U y = + +

( ) ( ) ( ) ( )< !$173$37 !32#32!! < 1; 1!< !32#32!! < 1!

$("";!#"11

37#332712#

R UR R UR

UR UR

ns

ESS ESS Edf Edf F ESS Edf

− − − −= =

= =

Regression ines for Kr%an 9 Rural are

E


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2929

E


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'ntroduce dummy varia%le for'ntroduce dummy varia%le for

structural changestructural change

;i= saving


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3131

Scatter plotScatter plot


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3232

S."S." && SSSS

RegReg

ErrorError

::

8888

AA1CB!A:>AA1CB!A:>

00CB1!8@:00CB1!8@:

otal otal 8@8@ BBAC1!1ACBBAC1!1AC

S."S." && SSSS

RegRegErrorError

008>8>

C??80!CAAC??80!CAA8:8>A!8BA8:8>A!8BA

otal otal 8@8@ BBAC1!1ACBBAC1!1AC

*estricted%odel

iβα Xi1 1

iU y = + +

Regression are di5erent for two time

( ) ( )( )

=!71"72#3#

"23#27

22

2#3117"

)222$(

2#3117"27;232$;

<

<==

−

−

=−−

=

URUR

UR RUR R

Edf ESS

Edf Edf ESS ESS F

)("!#"";"$71#2"1!18iii

DX X DY −++=i X Y "3;"$23!2

8 +=

iiiii U X D X DY +−++= )(

'odel*estricted>Un

1121 β β α α

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