Haruna Issahaku

DUMMY VARIABLES

BYHARUNA ISSAHAKU

Haruna Issahaku

Definition

• Dummy variables (DV) indicate the presence or absence of a ‘quality’ or an attribute such as male or female, black or white, north or south, etc.

• A dummy variable will take the value 1 or 0 according to whether or not the condition is present or absent for a particular observation.

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• 1 indicates the presence of the attribute• 0 indicates the absence of the attribute• Eg. 1 may represent female and 0 for male• DV are sometimes called categorical variables,

or qualitative variables or indicator variables• ANOVA MODELS: Used to test the statistical

significance of the relationship between a quantitative regressand and qualitative or dummy regressors

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• ANCOVA MODELS: regression models containing an admixture of quantitative and dummy regressors.– They provide a method of statistically controlling

for the effects of quantitative regressors called covariates or control variables in a model that includes both quantitative and qualitative regressors

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A single dummy independent variable

• Given the wage discrimination model

• Wage=hourly wage• D=1 for females and 0 for males• • is the difference in hourly wage between males and females

given the same level of education and the same error term u.• is the mean hourly wage for males (the base or benchmark

group)•

1o fWage B D B edu u

oB

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Why a single category is included in a single DV. equation

• Using two DV would introduce perfect colinearity because for example

• Female + male=1• Ie. Male is a perfect colinear function of female• Introducing DV for both male and female is the

simplest example of the so called dummy variable trap.

• It arises when too many DV describe a given number of groups

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Overcoming the DV trap

• One way of overcoming the DV trap is to drop the intercept and include the two categories

• Eg.1 .oWage B male female B educ u

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Some interpretations

• Given• Wage = 2.91 - 1.81female + 0.572educ.• Se= (0.12) (0.26) (0.049)• N=526 R-sqd=0.364• -males receive a mean wage of Ghc2.91 per hour• -on the average females receive Ghc1.81 less than

their male counterparts holding education constant.• -the average hourly wage of female is • 2.91-1.81=Ghc1.10

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Interpreting coefficients on DV when an explanatory v. is in log form

• Given the ff. model on the effects of training grants on hours of training by firms

• hrsemp=hours of training per employee at the firm level• Grant is a DV =1 if a firm received a job training grant

and 0 otherwise

2

ˆ 46.67 26.25 0.98log( ) 6.07 log( )

(43.21) (5.59) (3.54) (3.54)

105 0.237

hrsemp grant sales employ

se

n R

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• Sales=annual sales• Employ=total no. of employees of the firm• Grant coefficient of 26.25 means controlling

for sales and employment, firms that receive grants trained each worker on the average 26.25 hours more

• The coefficient on log sales is small and insignificant

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• The coefficient on log employ of -6.07 means• If a firm is 1% larger it trains its workers by

0.061 (ie. 6.07/100) hours less

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Interpreting DV when dependent v. is in log form

• 1. given the housing price equation

• Colonial is a DV. =1 if house is of a colonial style and 0 otherwise

• Raw interpretation: holding other factors constant the difference in log(price) of a house of a colonial style is 0.054

• Correct interpretation: a house of the colonial style is predicted to sell for about 5.4% more, holding other factors constant

• Ie. When the dependent v. is in log form the coefficient on the DV multiplied by 100 is interpreted as percentage difference in the dependent variable holding other factors constant.

ˆlog( ) 5.56 0.168( ) 0.707 log( ) 0.027 0.054price lotsize sqrft bdrms colonial

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2. Given a log hourly wage equation

• Using log(wage) as the dependent v. and adding quadratics in experience

• The coefficient on female implies for the same levels of experience and education women on the average earn 29.7% (ie. 100*0.297) less than men

• More accurate interpretation: a more accurate interpretation is obtained by using the formula

2log( ) 0.417 0.297 0.080 0.029exp 0.0005expwage female educ er er

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• 100*[exp.(B)-1]• Where B is the coefficient on the dummy

variable• Thus, 100*[exp.(-0.297)-1]=25.7• More accurately, on the average a woman’s

wage is 25.7% below a comparable man’s wage

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• Median hourly wage for males is calculated by taking the antilog of the intercept 0.417

• exp(0.417)=Ghc1.517• The median hourly wage of females• Exp[0.417+(-0.279)]=Ghc1.148

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ANOVA models with 2 DV.

• Model: hourly wage in relation to marital status and region of residence

• Y=hourly wage (Ghc)• =marital status 1=married 0=otherwise• =region of residence 1=south 0=otherwise

2 3

2

8.8148 0.997 1.6729

(0.4015) (0.4642) (0.4854)

(21.953) (2.3688) (0.0006) 0.0322

t i iY D D

se

t R

2D

3D

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• Which is the benchmark group?• Unmarried non-south residents• All comparisms are made wrt this group as ff:• -Unmarried non-south residents receive a

mean hourly wage of GhC8.81• -Those married on average receive Ghc1.10

more than the unmarried non-south residents

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• The mean hourly wage for the married is Ghc 9.91 (ie. 8.81+1.10)

• -the hourly wage of those from the south is lower by Ghc 1.67

• The mean hourly wage of those from the south is Ghc7.14 (ie. -1.67+8.81)

• All are statistically significant.

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Interaction effects using DV.

• Given

• Female is a dummy=1 for females; 0=otherwise• Married is a dummy 1=married 0=otherwise• The differential effect of being a married female

is 0.301• Married females earn 35.12% more than single

men with a mode of Ghc1.35

log( ) 0.321 0.110 0.213 0.301 .wage female male female married u

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• What will be the earnings for– 1. married men?– 2. single men?– 3. unmarried females?

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Uses of DV in applied econometric research

• Read Akoutsoyannis pp.281-284• 1. As proxies to qualitative variables• 2. As proxies for numerical factors• 3. Measuring shifts of a function over time• 4. Measuring the change in parameters over

time• 5. As proxies for the dependent variable• 6. For seasonal adjustment of time series

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Indicator vs effects coding

• Indicator coding: where the reference category is assigned zero across the set of DV.

• Effects coding: where the reference category is assigned a value of negative 1 across the set of DV.

• In indicator coding coefficients represent group deviations on the dependent variable from the reference group

• While in effect coding coefficients become group deviations on the dependent variable from the mean of the dependent variable across all groups

Haruna Issahaku

Haruna Issahaku

Haruna Issahaku