哑元变量 dummy variables

Mean salary of female accountant:

E(Y|X, D=0) = b1 + b3X

Expect a negative coefficient on D.
E(Y|X, D=1) = b1 + b2D + b3X
Including Both Categories

We cannot put in 2 dummies (1 for male and 1 for female)
Including Both Categories

Now D1 is simply 1- D2

whenever D2 is 1, D1 is 0; whenever D2 is 0, D1 is 1. whenever D1 is 1, D2 is 0; whenever D1 is 0, D2 is 1.
Accountant Exe the results:

Y = 35.20 + 10.25D t (43.82) (9.45)

Y is income in thousands D is a dummy variable taking on the value of 1 if male.
Add Quantitative Variable

Suppose we estimate this equation




Y = 25.20 + 8.25D + 1.25X t (32.85) (10.45) (15.11) Female salary is $25,200 assuming 0 years of experience. Male salary is $33,450 assuming 0 years of experience. One year of experience adds $1,250 to salary of either male or female accountants.
Education Categories

Education has 3 categories:

H.S. dropout, high school, college

Set up the dummies as:



Dhigh= 1 if high school graduate; 0 otherwise Dcollege = 1 if college graduate; 0 otherwise Our omitted category is less than high school

The level of the male salary is higher than the female, but the increase in salary as experience increases is the same for both male and female accountants

For a high school graduate it is:


For a college graduate it is:

E(Y|Dhigh = 0, Dcollege = 1, X) = b1 + b3 + b4X
Results

Suppose get the following results:
More than 2 Categories

Suppose examine expenditures on health care as a function of income and education.


Spending=f(income, education) Income is a continuous variable Have data only whether or not graduated from high school or college--not years of schooling.
Spending by Education

The mean value of spending for a less than high school individual is:

E(Y|Dhigh = 0, Dcollege = 0, X) = b1 + b4X E(Y|Dhigh = 1, Dcollege = 0, X) = b1 + b2 + b4X

Else will have perfect collinearity.

What if estimate model with female and male dummies.

D1 which is 1 if male, 0 if female D2 which is 1 if female, 0 if male
2. Model with Dummy Variable and Quantitative Variable
Add Quantitative Variable

Then model becomes:

Y = b1 + b2D + b3X + e



Y = accountants salary D = 1 if male, otherwise 0 X = years of experience
Add Quantitative Variable

The t value on the dummy variable indicates that, holding experience constant, the mean salaries of male and female accountants are different.
Chapter 9: Dummy Variables
Zongyi ZHANG
College of Economics and Business Administration
1. Introduction
Introduction



If have data, how to examine male-female difference in salary of accountants? If have data, how to examine male-female difference in salary of accountants after consideration of the difference of experiences? If have data, how to examine difference of management (directors of board) performance due to difference of education?

The estimated mean salary of female accountants is $35,200
Accountant Example


Estimated mean salary of male accountants is $45,450 Gender differential is statistically significant since b2 is significant.
Introduction

We can have dummy variables as explanatory variables along with quantitative variables or other dummy variables.
Accountant Example


Suppose examine male-female difference in salary of accountants Estimate:

Y=-5.3+2.7Dhigh+9.6Dcollege+.2X t (-7.9) (1.5) (2.3) (5.9) Holding education constant, as income increases by $1, average health care spending increases by 20 cents.
Estimating Model

Estimating model:

Y= b1+b2Dhigh +b3Dcollege+b4X+e


Y is annual health care spending X is annual income

Graphically, we have three regression lines with different intercepts, but the same slope:
Changing Omitted Category

Now make male the omitted category (the category equal to 0)

Y = b 1 + b 2D + b3X + e

Now D is 1 for female, 0 for male

Mean salary of male accountant:

Mean salary of female accountant:

E(Y|X, D=0) = b1 +b3X
Add Quantitative Variable

Mean salary of male accountant:

E(Y|X, D=1) = b1 + b2D + b3X

Graphically, we are assuming the same slope but different intercepts.

Y = b1 + b2D + e

Y is annual salary of accountant D is 1 if male and 0 if female.
Accountant Example


The mean salary of a female accountant is: E(Y|D=0) = b1 + b2(0) = b1 The mean salary of a male accountant is: E(Y|D=1) = b1 + b2 (1) = b1 + b2
Introduction

Sometimes the explanatory variables are qualitative (dummy variables)

Ex: gender, race, religion, region. Examples:


1 indicates male and 0 female. 1 indicates living in the South and 0 not living in the South. 1 indicates Republican and 0 Democrat.
Accountant Example




The intercept term gives the mean salary of a female accountant The dummy coefficient indicates how much the mean salary of a male differs from a female. The mean salary of males is the intercept plus dummy coefficient. Graph is a step function

D2 is 1- D1



D1 and D2 are perfectly collinear and so we can’t include both . Always leave out one of the dummy variable categories.
3. Model with Multiple Category Qualitative Variable and Quantitative Variable