Dummy Variable Regression

1

Dummy

Variable

Models

2

“Using Dummy Variables in

Wage Discrimination Cases”

Multiple Regression

Sandy: pages 603 - 613

Also read paper titled:

3Are Male Nurses Discriminated Against?

male nurses

0

female nurses

Years of experience, Xi

Wf

_4

^

Wm_

3^

~m

W 3

~

Wf

~

4

~

~

adjusted for

experience

not adjusted

for experience

o

ooo o

oo

o

o

oo

o

++ +

+++

+

+

+

+

+

+

+

+

+

++

+

+++++

++

o

oo

~

4

I. Dummy Variables -

Adjusting the intercept.

Adjusting the slope.

Adjusting both intercept and slope.

5Intercept Dummy Variables

Dummy variables are binary (0,1)

Dt = 1 if red car, Dt = 0 otherwise.

yt = 1 + 2Xt + 3Dt + et

yt = speed of car in miles per hour

Xt = age of car in years

Police: red cars travel faster.H0: 3 = 0H1: 3 > 0

6yt = 1 + 2Xt + 3Dt + et

red cars: yt = (1 + 3) + 2Xt + et other cars: yt = 1 + 2Xt + et

yt

Xt

milesper hour

age in years0

1 + 3

12

2

red cars

other cars

7Slope Dummy Variables

yt = 1 + 2Xt + 3DtXt + et

yt = 1 + (2 + 3)Xt + et

yt = 1 + 2Xt + et

yt

Xt

valueofporfolio

years0

2 + 3

12

stocks

bonds

Stock portfolio: Dt = 1 Bond portfolio: Dt = 0

1 = initial

investment

8Different Intercepts & Slopes

yt = 1 + 2Xt + 3Dt + 4DtXt + et

yt = (1 + 3) + (2 + 4)Xt + et

yt = 1 + 2Xt + et

yt

Xt

harvestweightof corn

rainfall

2 + 4

12

“miracle”

regular

“miracle” seed: Dt = 1 regular seed: Dt = 0

1 + 3

9yt = 1 + 2 Xt + 3 Dt + et

21+ 3

2

1

yt

Xt

Men

Women

0

yt = 1 + 2 Xt + et

For men Dt = 1. For women Dt = 0.

years of experience

yt = (1+ 3) + 2 Xt + et

wagerate

H0: 3 = 0

H1: 3 > 0 .

. Testing fordiscriminationin starting wage

10yt = 1 + 5 Xt + 6 Dt Xt + et

5

5 +6

1

yt

Xt

Men

Women

0

yt = 1 + (5 +6 )Xt + et

yt = 1 + 5 Xt + et

For men Dt = 1.

For women Dt = 0.

Men and women have the same starting wage, 1 , but their wage ratesincrease at different rates (diff.= 6 ).

6 > means that men’s wage rates areincreasing faster than women's wage rates.

years of experience

wagerate

11yt = 1 + 2 Xt + 3 Dt + 4 Dt Xt + et

1 + 3

1

2

2 + 4

yt

Xt

Men

Women

0

yt = (1 + 3) + (2 + 4) Xt + et

yt = 1 + 2 Xt + et

Women are given a higher starting wage, 1 , while men get the lower starting wage, 1 + 3 ,(3 < 0 ). But, men get a faster rate of increasein their wages, 2 + 4 , which is higher than therate of increase for women, 2 , (since 4 > 0 ).

years of experience

An Ineffective Affirmative Action Plan

women are startedat a higher wage.

Note:(3 < 0 )

wagerate

12Testing Qualitative Effects

1. Test for differences in intercept.

2. Test for differences in slope.

3. Test for differences in both

intercept and slope.

13

H0: vs1:

H0: vs1:

Yt 12Xt3Dt

4Dt Xt

b3

Est. Var b3 ˜ t n 4

b4

Est. Var b4 ˜ t n 4

men: Dt = 1 ; women: Dt = 0

Testing fordiscrimination instarting wage.

Testing fordiscrimination inwage increases.

intercept

slope

et

14Why NOW wants one-sided test andChauvinist Industries wants two-

sided.

15Are Two Regressions Equal?

yt = 1 + 2 Xt + 3 Dt + 4 Dt Xt + et

variations of “The Chow Test”

I. Assuming equal variances (pooling):

men: Dt = 1 ; women: Dt = 0

Ho: 3 = 4 = 0 vs. H1: otherwise

yt = wage rate

This model assumes equal wage rate variance.

Xt = years of experience

16Testing Ho: H1 : otherwise

and

SSE R yt b 1 b 2 X t 2

t 1

T

SSE U yt b1bXt bDt b Dt Xt2

t1

T

SSER SSEU 2

SSEU T 4 F T 4

intercept and slope

17

yt = 1 + 2 Xt + et

II. Allowing for unequal variances:

ytm = 1 + 2 Xtm + etm

ytw = 1 + 2 Xtw + etw

Everyone:

Men only:Women only:

SSER

Forcing men and women to have same 1, 2.

Allowing men and women to be different.SSEm

SSEw

where SSEU = SSEm + SSEw

F =(SSER SSEU)/J

SSEU /(TK)

J = # restrictions

K=unrestricted coefs.

(running three regressions)

J = 2 K = 4

18 Polynomial Terms

yt = 1 + 2 X t + 3 X2

t + 4 X3

t + et

Linear in parameters but nonlinear in variables:

yt = income; Xt = agePolynomial Regression

yt

X tPeople retire at different ages or not at all.

9020 30 40 50 60 8070

19

yt = 1 + 2 X t + 3 X2

t + 4 X3

t + et

yt = income; Xt = age

Polynomial Regression

Rate income is changing as we age:yt

Xt

= 2 + 2 3 X t + 3 4 X

2t

Slope changes as X t changes.

20 Continuous Interaction

yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et

Exam grade = f(sleep:Zt , study time:Bt)

Sleep and study time do not act independently.

More study time will be more effective when combined with more sleep and less effective when combined with less sleep.

21

Your mind sortsthings out whileyou sleep (when you have things to sort out.)

yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et


yt

Bt

= 2 + 4 Zt

Your studying is more effectivewith more sleep.

yt

Zt

= 2 + 4 Bt

continuous interaction

22

yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et


If Zt + Bt = 24 hours, then Bt = (24 Zt)

yt = 1+ 2 Zt +3(24 Zt) +4 Zt (24 Zt) + et

yt = (1+24 3) + (23+24 4)Zt 4Z2

t + et

yt = 1 + 2 Zt + 3 Z2

t + et

Sleep needed to maximize your exam grade:yt

Zt

= 2 + 23 Zt = 0where 2 > 0 and 3 < 0

2

3

Zt =

23

Multicollinearity

Correlation among the“independent” variables.

Note: They are independent of the error term, and not of one another.

24 Let yi represent the ith person's wagerate and Xi represent their monthsof work experience in the equation:

yi = b1 + b2 Xi + ei (1)

b1 = intercept (starting wage)

b2 = increase in the person's wage for each additional month of work experience.

ei = error term with mean zero and estimated variance s2.

25

yi = b1 + b2 Xi + b3 Mi + b4 Fi + ei (2)

Fi = 1 if female Fi = 0 if male.

Mi = 1 if male Mi = 0 if female.

26

yi = b1 + b2 Xi + b3 Mi + b4 Fi + ei (2)

Unfortunately this equation contains

an underidentified set of parameters

(b1, b3, and b4) and cannot be estimated

without some restriction

on the coefficients.

27

To see this point, separate out the men's equation implied by equation (2) from the women's equation.

For the men's equation Mi =1 and Fi =0.

For men, equation (2) becomes:

yi = (b1 + b3) + b2 Xi + ei (3)

yi = b1 + b2 Xi + b3 Mi + b4 Fi + ei (2)

28

For women, Mi =0 and Fi =1.

For women, equation (2) becomes:

yi = (b1 + b4) + b2 Xi + ei (4)

29 Unfortunately, although we get estimates

of the intercepts (b1 + b3) and (b1 + b4),

the value of b1 cannot be separated

from the values of b3 and b4.

Some restriction is needed

to achieve identification

of b1, b3 and b4.

30

One such restriction is b1 = 0.

We can drop the original intercept term,

b1, since men and women already

have their own intercept terms,

b3 and b4, respectively.

31 Underidentification of equation (2) can also be expressed in matrix terms.

First, rewrite equation (2) putting the explanatory variables in a row vector multiplied by the corresponding column

vector of their respective coefficients:

y i1X iM iF i 2

3

4

i5

1

32 This only represents the ith observation where i = 1, ..., n.

To represent the entire set of n observations at once, we need to "pull the window shade down" as follows:

y1

y2

M

yn

1 X1 M1 F1

1 X2 M2 F2

M M M M

1 Xn Mn Fn

1

2

3

4

1

2

M

n

(6)

33

Equation (6) presents us with an X matrix whose first column (the column of ones) is an exact linear combination of the last two columns (the M and F columns).

Since Mi is always zero when Fi is equal to one and Mi is always one when Fi is equal to zero, then it always holds that Mi + Fi = 1.

Therefore, the first column is equal to the sum of the last two columns.

34Since Mi is always zero when Fi is equal to one and Mi is always one when Fi is equal to zero, then it always holds that Mi + Fi = 1.

1

1

M

1

M1

M2

M

Mn

F1

F2

M

Fn

(9)

35 Equation (6) and, therefore,equation (2), represent a case of perfect multicollinearity. This means that a restriction must be introduced that drops one of these columns out of the regression.

One such restriction is b1 = 0, which means dropping the originalintercept out of the regression model to provide the following reduced model:

yi = b2 Xi + b3 Mi + b4 Fi + ei (10)

Now men and women have separate interceptsand no common intercept is necessary.

36yi = b2 Xi + b3 Mi + b4 Fi + ei

b2b3

b2

b4

yi

Xi

Male

Female

0

yi = b3 + b2 Xi + ei


For males Mi = 1 and Fi = 0.For females Mi = 0 and Fi = 1.

Males and females have differentstarting salaries , b3 > b4 , but theirsalaries increase at the same rate, b2.

37yi = b2 Xi + b3 Mi + b4 Fi + ei

b2b3

b2

b4

yi

Xi

Male

Female

0





years of experience

38yi = b1 + b5 Mi Xi + b6 Fi Xi + ei

b6

b5

b1

yi

Xi

Male

Female

0




Males and Females have the same starting salary b1, but their salariesincrease at different rates ( b5 vs. b6 ).

b5 > b6 means that men salaries areincreasing faster than women's salaries.

years of experience

39yi = b3 Mi + b4 Fi + b5 Mi Xi + b6 Fi Xi + ei

b3b4


b6

b5

yi

Xi

Male

Female

0



Females start with a higher starting salary, b4, while men get the lower starting salary, b3.But, men get a faster rate of increase intheir salaries, b5, which is higher than therate of increase for females, b6. ( b5 > b6 ).

years of experience

Chauvinist Industries Affirmative Action Plan

40yi = b2 Xi + b3 Mi + b4 Fi + ei

b2b3

b2

b4

yi

Xi

Male

Female

0





Back to our basic model:

years of experience

41 Since under our null hypothesis

the raw score test statistic: has a mean and a variance, we can standardize by subtracting the mean (zero) and dividing by the standard deviation (square root of the variance) to get the standardized test statistic:

β 3 – β4 = 0

b3 – b4

Var (b3 – b4 )

b3 – b4

42To test the null hypothesis:

H0: − =wecoulduse

thestandardizedteststatistic:

Z (b b ) 0

Var(b b )~ (0,1)

43If the variance of the y i , 2 ,

is unknown , then Var (b3 b4 )

is also unknown and must be

estimated from the expression :

Est.Var(b3b

4)

Est.Var(b3) Est.Var(b

4) 2 Est.Cov(b

3,b

4)

44 Use the sample variance as anestimator of the population variance:

Use s2 as an unbiased estimator of

where: s =(yi −ˆyi)

i=1

n

∑(n−k −1)

wherenisthenumberofobservations

andkisthenumberofindependent

variables.

45 The values for the following expressionare obtained in practice from the diagonaland off-diagonal elements of the estimated variance-covariance matrix:

Est . Var (b3

b4)

Est .Var (b3) Est . Var (b

4) 2 Est . Cov (b

3,b

4)

46

yi = b1 + b2 Xi + b3 Mi

b2(b1 + b3)

b2

b1

yi

Xi

Male

Female

0

yi = ( b1 + b3 ) + b2 Xi

yi = b1 + b2 Xi

Males and females have differentstarting salaries , b3 > 0 , but theirsalaries increase at the same rate, b2.

years of experience

Alternative: make women the default group

^

^

^

47

yi = b1 + b2 Xi + b3 Mi + b4 Di

yi = (b1 + b3 + b4) + b2 Xi

yi = (b1 + b4) + b2 Xi

yi = (b1 + b3) + b2 Xi

yi = b1 + b2 Xi

characteristic dummy variables:

male college grad:

female college grad:

male not a grad:

female not a grad:

^

^

^

^

^

48

years of experience0 X i

M-D (male-degree)

F-D (female-degree)

M-N (male-no degree)

F-N (female-no degree)

yiwage rate

very restrictive assumption

yi = b1 + b2 Xi + b3 Mi + b4 Di

b1b1+b3

b1+b4b1+b3+b4

very rigid !!!

^

49

Creating

Composite

Dummy Variables

( vs. characteristic dummy variables )

50

Job:

Gender:

Karnaugh map for gender vs. status of job:

S I

M 15 25 40

F 13 27 40

28 52 80

S = supervisor I = individual

men:

women:

51Occupation vs. Job vs. Gender

Gender:

Occupation:

Job:

C T US I S I S I

M 2 4 3 5 10 16 40

F 1 6 0 7 12 14 40

3 10 3 12 22 30 80

C = ComputerT = Other TechnicalU = Untechnical

52 Karnaugh Map for Occupation,

Job Status, Gender, and Degree Status:

Degree

NoDegree

C T US I S I S I

D M 1 3 2 5 6 13 30

F 0 3 0 6 7 8 24

N M 1 1 1 0 4 3 10

F 1 3 0 1 5 6 16

3 10 3 12 22 30 80

53composite dummy variables:

This defines combined( instead of separate )general characteristics.

yi = b1 + b2 Xi + b3 MNi + b4 FDi + b5 MDi

years of experience0 X i

M-D (male-degree)

F-D (female-degree)

M-N (male-no degree)

F-N (female-no degree)

yiwage rate

b1

b1 + b3

b1 + b4

b1 + b5

^

54

Multiple Regression Analysis

value of

residential property

( buying a home )

55Ai = bathrooms Xi = sq. ft. living space

H0: vs. H1:

H0: vs. H1:

ˆ Y i b1 b2Xi b3A i b4Ai Xi

b3Est. Varb3 ˜ tn 4

b4Est. Varb4 ˜ tn 4

56 Testing Ho: H1 : otherwise

and

( SSER

− SSEU

) / 2

SSEU

/ ( n − 4 )

∼ Fn − 4

2

SSE R yi b 1 b 2 X i 2

i 1

n

SSEU yi b1bXi bA ibA iX i2

i1

n

57Sale of House with Bed and Bath Dummies

800 0 0 0 10.0001000 0 0 1 20.0001200 1 0 0 30.0001500 1 0 0 40.0001800 1 0 1 50.0002000 1 0 1 60.0002200 0 1 0 70.000

2500 0 1 0 80.0003000 0 1 1 90.000

3500 0 1 1 100.000

PRICE = f ( SQFEET, D2BED, B3BED, A2BATH )

I. II. III. IV. PRICE (thousands)

I. SQFEET = square feet of living space II. D2BED = dummy=1 if two-bedroom house III. D3BED = dummy=1 if three-bedroom house IV. A2BATH = dummy=1 if two-bathroom house

58PRICE = f ( SQFEET, D2BED, B3BED, A2BATH )

Sale of House with Bed and Bath Dummies

ANALYSIS OF VARIANCE

SOURCE SUM-OF-SQUARES DF MEAN-SQ F-RATIO P

REGRESSION 8191.943 4 2047.986 176.378 0.000 RESIDUAL 58.057 5 11.611

DURBIN-WATSON D STATISTIC: 2.216FIRST ORDER AUTOCORRELATION COEFF: - 0.153

DEP VAR: PRICE N: 10 MULTIPLE R: 0.996 SQUARED MULTIPLE R: 0.993

ADJUSTED SQUARED MULTIPLE R: 0.987

STD ERROR OF ESTIMATE: 3.40

59PRICE = f ( SQFEET, D2BED, B3BED, A2BATH )

Sale of House with Bed and Bath Dummies

DEP VAR: PRICE N: 10 MULTIPLE R: 0.996

SQUARED MULTIPLE R: 0.993

ADJUSTED SQUARED MULTIPLE R: 0.987 STD ERROR OF ESTIMATE: 3.40 VARIABLE COEFF STD ERR T P(2-TAIL)

INTERCEPT - 6.482 4.112 -1.576 0.176 SQFEET 0.021 0.005 3.958 0.011 D2BED 14.662 4.871 3.010 0.030 D3BED 29.803 10.575 2.818 0.037 A2BATH 4.883 3.953 1.235 0.272 ( for 1,000 square feet: 21 - 6.482 = 14.518 or $14,518 )

60 VARIABLE COEFF STD ERR T P(2-TAIL)

INTERCEPT - 6.482 4.112 -1.576 0.176 SQFEET 0.021 0.005 3.958 0.011 D2BED 14.662 4.871 3.010 0.030 D3BED 29.803 10.575 2.818 0.037 A2BATH 4.883 3.953 1.235 0.272

for 1,000 square feet: 21 - 6.482 = 14.518 or $14,518

add a bathroom:

$14,518 4,883

$19,401

add a bedroom:

$14,518 14,662

$29,180

add 2 bedrooms:

$14,518 29,803

$44,321

add bath and 2 bedrooms: 14,518 + 4,883 + 29,803 = $49,204

Regression Analysis of Sale of Residential Property

61Sales Value of Residential Property

y = sales value of the property (dollars)X = square feet of living spaceD1=dummy vble for one bedroom homeD2=dummy vble for two bedroom homeD3=dummy vble for three bedroom homeA1=dummy vble for one bathroom homeA2=dummy vble for two bathroom home

For a one-bedroom, one-bathroom home, such that D2=0, D3=0, and A2=0, we have:

yib

1b

2X

ib

3D2

ib

4D3

ib

5A2i

^

yib

1 b

2X

i 1 bedroom , 1 bathroom^


For a 2-bedroom, 1-bathroom home,

we have D2=1, D3=0, and A2=0

^

^

yib

1b

2X

ib

3D2

ib

4D3

ib

5A2 i

yi(b

1 b

3) b

2X

i 2 bedroom, 1 bathroom


For a 1-bedroom, 2-bathroom home,

we have D2=0, D3=0, and A2=1

^

^

yib

1 b

2X

i b

3D2

i b

4D3

i b

5A2

i

yi(b

1 b

5) b

2X

i 1 bedroom, 2 bathroom


For a 2-bedroom, 2-bathroom home, we have D2=1, D3=0, and A2=1

yib

1 b

2X

i b

3D2

i b

4D3

i b

5A2

i

^

yi (b

1 b

3 b

5) b

2X

i 2 bedroom , 2 bathroom

^

yi (b

1 b

4 b

5) b

2X


yi (b

1 b

4) b

2X


65

square feet of living space

0 X i

House Sales Model with Restricted Intercepts

b1bb

D2-A2 (two bed, two bath)

b1b

D2-A1 (two bed, one bath)

b1b

D1-A2 (one bed, two bath)

b1

D1-A1 (one bed,one bath)

yi

selling price

b1bb

D3-A2 (three bed, two bath)

b1b

D3-A1 (three bed, one bath)

b

yib

1 b

2X

i b

3D2

i b

4D3

i b

5A2

i

^

^

Rigid !!!

66

Creating

Composite

Dummy Variables


67

Bath-rooms

How do we create composite dummy variables?

Need to account for the interactioneffect betweenbathrooms andbedrooms. 1 2 3

1 6 8 26 40

2 7 7 26 40

13 15 52 80

Bedrooms

68 Composite dummy variables are created for each nonempty cell.

Create six composite dummy variables:

D1A1=1 if one bed and one bath, or D1A1= 0

D1A2=1 if one bed and two bath, or D1A2= 0

D2A1=1 if two bed and one bath, or D2A1= 0

D2A2=1 if two bed and two bath, or D2A2= 0

D3A1=1 if three bed and one bath, or D3A1= 0

D3A2=1 if three bed and two bath, or D3A2= 0


y = sales value of the property (dollars)X = square feet of living spaceD1A1 = interaction one-bed & one-bathD1A2 = interaction one-bed & two-bathD2A1 = interaction two-bed & one-bathD2A2 = interaction two-bed & two-bathD3A1 = interaction three-bed & one-bathD3A2 = interaction three-bed & two-bath

yib

1 b

2X

i b

3D1A2

i b

4D2A1i

b5D2A2 i

^

b6D3A1i

b7D3A2 i

70This one equation with all these dummyvariables actually is representing sixequations. You must substitute in foreach of the dummy variables to generatethe six equations that are implied by this one dummy variable equation.

For a one-bedroom, one-bathroom home, Since D1A1 = 1, while the others are zero:

yib

1 b

2X

i 1 bedroom , 1 bathroom

^

yib

1 b

2X

i b

3D1A2

i b

4D2A1i

b5D2A2 i

^

b6D3A1i

b7D3A2 i

71


0 X i

House Sales Model with Unrestricted Intercepts




b1


yi

selling priceD3-A2 (three bed, two bath)


b

72

one-bedroom, two-bathroom

D1A2 =1, while the others are zero:

now graph it ! =======>

yi(

1 b

3) b

2X

i 1 bedroom, 2 bathroom^

yib

1 b

2X

i b

3D1A2

i b

4D2A1i

b5D2A2 i

^

b6D3A1i

b7D3A2 i

b

73


0 X i



b1b



b1


yi

selling price D3-A2 (three bed, two bath)


74

two-bedroom, one-bathroom


yi

(b1

b4

) b2X


yib

1 b

2X

i b

3D1A2

i b

4D2A1i

b5D2A2 i

^

b6D3A1i

b7D3A2 i


75


0 X i



b1b


b1b


b1


yi

selling price D3-A2 (three bed, two bath)


76

two-bedroom, two-bathroom


yi

(b1

b5

) b2X


yib

1 b

2X

i b

3D1A2

i b

4D2A1i

b5D2A2 i

^

b6D3A1i

b7D3A2 i


77


0 X i


b1b


b1b


b1b


b1


yi

selling priceD3-A2 (three bed, two bath)


78


0 X i


b1b


b1b


b1b


b1


yi

selling price

b1b

D3-A2 (three bed, two bath)

b1b


79

Creating

Composite

Dummy Variables


80

Bath-rooms

How do we create composite dummy variables?

Need to account for the interactioneffect betweenbathrooms andbedrooms. 1 2 3

1 6 8 26 40

2 7 7 26 40

13 15 52 80

Bedrooms

81Bedrooms vs. Baths vs. Garage

Baths

Bedrooms

Cars inGarage:

1 2 3

1 2 1 2 1 2

1 2 4 3 5 10 16 40

2 1 6 0 7 12 14 40

3 10 3 12 22 30 80

82Karnaugh Map for Bedrooms,

Baths, Garage, and School:

Adams

SaintJoseph

1 2 31 2 1 2 1 2

A1 1 3 2 5 6 13 30

2 0 3 0 6 7 8 24

J1 1 1 1 0 4 3 10

2 1 3 0 1 5 6 16

3 10 3 12 22 30 80

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Dummy Variable Regression

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