Is There Evidence of Wage Discrimination?
◼
Three Seton Hall professors recently learned in a court
decision that they could pursue their lawsuit alleging the
University paid higher salaries to younger instructors and
male professors.
◼
Mary Schweitzer works in human resources at another
college and has been asked by the college to test for
age and gender discrimination in salaries.
◼
She gathers data on 42 professors, including the salary,
experience, gender, and age of each.
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Is There Evidence of Wage Discrimination?
◼
Using this data set, Mary hopes to:
1.
Test whether salary differs by a fixed amount between
males and females.
2.
Determine whether there is evidence of age
discrimination in salaries.
3.
Determine if the salary difference between males and
females increases with experience.
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17.1 Dummy Variables
LO 17.1 Use dummy variables to capture a shift of the intercept.
◼
In previous chapters, all the variables used in regression
applications have been quantitative.
◼
In empirical work it is common to have some variables
that are qualitative: the values represent categories that
may have no implied ordering.
◼
We can include these factors in a regression through the
use of dummy variables.
◼
A dummy variable for a qualitative variable with two
categories assigns a value of 1 for one of the categories
and a value of 0 for the other.
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LO 17.1
◼
Variables with Two Categories
For example, suppose we are interested in determining
the impact of gender on salary.
◼
We might first define a dummy variable d that has the
following structure:
◼
Let
d=1
if
gender = “female”
◼
and
d=0
if
gender = “male.”
◼
This allows us to include a measure for gender in a
regression model and quantify the impact of gender on
salary.
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LO 17.1
Regression with a Dummy Variable
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LO 17.1
Regression with a Dummy Variable
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LO 17.1
Regression with a Dummy Variable
Graphically, we can see how the dummy variable shifts the
intercept of the regression line.
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LO 17.1
Salaries, Gender, and Age
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LO 17.1
Estimation Results
◼
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Testing the Significance of Dummy Variables
LO 17.2 Test for differences between the categories of a qualitative variable.
◼
The statistical tests discussed in Chapter 15
remain valid for dummy variables as well.
◼
We can perform a t-test for individual
significance, form a confidence interval using the
parameter estimate and its standard error, and
conduct a partial F test for joint significance.
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LO 17.2
Example 17.2
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Multiple Categories
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Multiple Categories
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LO 17.2
Multiple Categories
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Avoiding the Dummy Variable Trap
◼
Given the intercept term, we exclude one of the dummy
variables from the regression,
◼
where the excluded variable represents the reference
category against which the others are assessed.
◼
If we included as many dummy variables as categories,
this would create perfect multicollinearity in the data, and
such a model cannot be estimated.
◼
So, we include one less dummy variable than the
number of categories of the qualitative variable.
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17.2 Interactions with Dummy Variables
LO 17.3 Use dummy variables to capture a shift of the intercept and/or slope.
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LO 17.3
Modeling Interaction
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LO 17.3
Shifts in the Intercept and the Slope
Graphically, we can see how both the intercept and the
slope might be impacted.
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LO 17.3
Testing for Significance
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LO 17.3
◼
Example 17.4
Our introductory case, is about the impact of gender on
salary. Further “Does additional experience get a higher
reward for one gender over the other?”
◼
Since age was not significant, we shall consider three
models, one with a dummy variable for gender, one with an
interaction variable between gender and experience, and
one with both a dummy variable and an interaction variable.
◼
As before, we keep experience as a quantitative
explanatory variable.
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◼
17-22
◼
17-23
◼
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LO 17.3
Predicted Salaries
◼
The interaction term allows for male professors to have a
different slope coefficient than female professors.
◼
Conceptually, experience impacts the salary of each
gender differently.
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17.3: Binary Choice Models
LO 17.4 Use a linear probability model to estimate a binary response variable.
◼
So far, we have been considering models where dummy
variables are used as explanatory variables.
◼
There are, however, many applications where the
variable of interest, the response variable, is binary.
◼
Consumer choice literature has many applications
including whether to buy a house, join a health club, or
go to graduate school.
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LO 17.4
The Linear Probability Model
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LO 17.4
Weakness of LPM
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Approval of Loan application
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◼
Here 0.0188 means a 1 percent increase in the down
payment will increase the probability by 0.0118.
◼
Similarly, a 1 percent increase in the income-to-loan ratio
will increase the probability of getting a loan by 0.0258.
◼
If the DP = 30%, IL = 30%, then the predicted
probability= 1.0338, which does not make sense.
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The Logit Model
◼
To address the problem of the LPM that the predicted
probabilities may be negative or greater than 1,
◼
we consider an alternative called the logistic model,
often referred to as a logit model.
◼
A logit model uses a nonlinear regression function that
ensures that the result is always in the interval [0,1].
◼
But interpreting the coefficients becomes more
complicated
◼
and estimation cannot be done by OLS.
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LO 17.5
Logistic Regression
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LO 17.5
Logit versus Linear Probability Model
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LO 17.5
Example 17.6
◼
An educator wants to determine if a student’s interest in
science is linked with the student’s GPA.
◼
She uses Minitab to estimate a logit model where a
student’s choice of field (1=science, 0=other) is predicted
by GPA.
◼
With a p-value of 0.0012, GPA is indeed a significant
factor in predicting whether a student chooses science.
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LO 17.5
Predicted Field Choice
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LO 17.5
Example 17.7
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LO 17.5
Prediction Comparison
◼
Compared to the linear probability model, the logit model does
not predict probabilities less than zero or greater than one.
◼
Therefore, whenever possible, it is generally preferable to use
the logit model rather than the linear probability model.
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Chapter 17 Learning Objectives (LOs)
LO 17.1: Use dummy variables to capture a shift of the
intercept.
LO 17.2: Test for differences between the categories of a
qualitative variable.
LO 17.3: Use dummy variables to capture a shift of the
intercept and/or slope.
LO 17.4: Use a linear probability model to estimate a binary
response variable.
LO 17.5: Interpret the results from a logit model.
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