DummyVariable Regression Model

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Dummy-Variable Regression Model
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Multiple Regression Models
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Dummy-Variable Regression Model

• Involves categorical X variable with 2 or more
  levels
• e.g., male-female, college-no college etc.
• or firms or states or cities
• Each level is coded 0 or 1
• Assumes only intercept is different
• Slopes are constant across categories
• The number of dummy variables that are included
  is 1- of levels

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Dummy-Variable Regression Model Example Coding

• Gender (2 levels) Male1 Female0 for variable
  MALE
• Marital Status (3 levels - requires 2 dummies)
• MARRIED Single0 Divorced0 Married1
• DIVORCED Single0 Divorced1 Married0

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Interpreting Dummy-Variable Model Equation
?
b
X
Y
b
?
?
Given

i
i
0
2
2

Y
?
Starting s
alary of c
ollege gra
d'
s

0
i
f Male
X
?
2
1
if Female
b0 mean Y for men since for each man
Yb0b2(0) b2 difference of means between men
and women since for women Yb0b2(1). b0b2
mean Y for women
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Comparison to other techniques

• This is identical to a t-test for the difference
  of means. We test b20 to test if there is a
  significant difference of means.
• This is identical to a one-way ANOVA for a
  difference of means.

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Dummy-Variable Model Relationships
Y
Means for males and females
Females
b0 b2
b0
Males
0
X1
0
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Interpreting Dummy-Variable Model Equation
?
Y
b
b
X
b
X
?
?
?
Given

i
i
i
0
1
1
2
2

Y
?
Starting s
alary of c
ollege gra
d'
s

X
?
GPA
1
0
i
f Male
X
?
2
1
if Female
Males (
)
X
?
0
2
?
Y
b
b
X
b
b
b
X
?
?
?
?
?
0
i
i
i
0
1
1
2
0
1
1
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Interpreting Dummy-Variable Model Equation
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Dummy-Variable Model Relationships
Y
Same slopes b1
Females
b0 b2
b0
Males
0
X1
0
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Dummy-Variable Model Example
Same slopes
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Interpretation

• The difference in mean output between men and
  women is 7, holding constant GPA.
• When there are more than two groups, the
  interpretation of the coefficient is always the
  difference of means between that group and the
  EXCLUDED GROUP.

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How many dummy variables do you need?

• To compare union workers and nonunion workers?
• To compare whites, blacks, hispanics and asians?
• To compare months of the year?

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EXAMPLE
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EXAMPLE
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Interaction Regression Model
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Multiple Regression Models
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Interaction Regression Model

• Hypothesizes interaction between pairs of X
  variables
• Response to one X variable varies at different
  levels of another X variable
• Contains two-way cross product terms
• Can be combined with other models
• e.g., dummy variable model

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Effect of Interaction

• Given
• Without interaction term, effect of X1 on Y is
  measured by ?1
• With interaction term, effect of X1 onY is
  measured by ?1 ?3X2
• Effect changes as X2i increases

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Interaction Example
Y 1 2X1 3X2 4X1X2
Y
Y 1 2X1 3(1) 4X1(1) 4 6X1
12
8
Y 1 2X1 3(0) 4X1(0) 1 2X1
4
0
X1
0
1
0.5
1.5
Effect (slope) of X1 on Y does depend on X2 value
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Interaction Regression Model Worksheet
Multiply X1 by X2 to get X1X2. Run regression
with Y, X1, X2 , X1X2
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Interpretation when there are 3levels
a Mean Y for a single female (MALE,MARRIED,DIVOR
CED0) b1 Difference in means between males and
females (ab1mean Y for single males) b2
Difference in means between single and married
(holding gender constant) b3 Difference in means
between divorced and single b2-b3Difference in
means between married and divorced
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Interpretation when there are 3levels

• It is possible to interact the dummy variables.
  This can give an identical result as a 2-way
  ANOVA.
• In this example, this would allow the effect of
  marital status to vary with gender.

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Interpretation when there are 3levels

• MALE0 if female and 1 if male

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Interpretation when there are 3levels

• MALE0 if female and 1 if male
• MARRIED1 if married 0 if divorced or single
• DIVORCED1 if divorced 0 if single or married
• MALEMARRIED1 if male married 0 otherwise
  (MALE times MARRIED)
• MALEDIVORCED1 if male divorced 0
  otherwise(MALE times DIVORCED)

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(No Transcript)
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Interpreting Results
Difference

• FEMALE
• Single
• Married
• Divorced

• MALE
• Single
• Married
• Divorced

Main Effects MALE (MARRIED and
DIVORCED) Interaction Effects MALEMARRIED and
MALEDIVORCED
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Interpreting results

• Testing for interaction Must do F-test of joint
  hypothesis that
• EXAMPLE

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Polynomial (Curvilinear) Regression Model
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Multiple Regression Models
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Curvilinear Regression Model

• Relationship between 1 response variable and 2 or
  more explanatory variable is a polynomial
  function
• Useful when scatter diagram indicates non-linear
  relationship
• Curvilinear model
• The second explanatory variable is the square of
  the 1st.

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Curvilinear Regression Model
Curvilinear models may be considered when scatter
diagram takes on the following shapes
Y
Y
Y
Y
X1
X1
X1
X1
?2 gt 0
?2 gt 0
?2 lt 0
?2 lt 0
?2 the coefficient of the quadratic term
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Testing for Significance Curvilinear Model

• Testing for Overall Relationship
• Similar to test for linear model
• F test statistic
• Testing the Curvilinear Effect
• Compare curvilinear model
• with the linear model

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Testing for Significance Curvilinear Model

• May require testing a portion of the model (e.g.
  the linear and squared terms) when there are
  other variables in the model
• Here we must test to test for the
  significance of X1 - an F-test for these two
  variables

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Inherently Linear Models

• Non-linear models that can be expressed in linear
  form
• Can be estimated by LS in linear form
• Require data transformation
• Multiplicative model example

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Using Transformations

• Requires Data Transformation
• Either or Both Independent and Dependent
  Variables May be Transformed
• Can be based on theory, logic or scatter diagrams

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Square Root Transformation
?1 gt 0
Similarly for X2
?1 lt 0
Transforms one of above model to one that appears
linear. Often used to overcome heteroscedasticity.
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Logarithmic Transformation
?1 gt 0
Similarly for X2
?1 lt 0
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Exponential Transformation
Original Model
?1 gt 0
Similarly for X2
?1 lt 0
Transformed into
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Interpretation of coefficients

• The dependent variable is logged.
• The coefficient on the independent variable can
  be approximately interpreted as a 1 unit change
  in X leads to a b percentage change in Y.
• The independent variable is logged.
• The coefficient on the independent variable can
  be approximately interpreted as a 100 percent
  change in X leads to a b unit change in Y.

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Interpretation of coefficients

• Both dependent and independent variables are
  logged.
• The coefficient on the independent variable can
  be approximately interpreted as a 1 percent
  change in X leads to a b percentage change in Y.
  Therefore b is the elasticity of Y with respect
  to a change in X.

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Income and Experience Scatter Plot
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Income and Experience Linear

• Linear Model

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Income and Experience Log Independent Variable

• Log independent variable

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Income and Experience Income Logged

• Log(Y)

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Income and Experience Double Log

• Double Log - Elasticity Model (Note LFEXP is
  already logged in this example)

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Income and Experience Quadratic

• Quadratic

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Income and Experience Log plus Quadratic

• Log(Y) Quadratic

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Income and Experience All Specifications

• Many specifications

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Standardized and Unstandardized

• Many disciplines report ONLY standardized
  coefficients
• The usual coefficients are then referred to as
  unstandardized coefficients
• The standardized coefficient are often referred
  to as beta weights
• The t-tests for significance of the slopes are
  identical for either of these two.

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Interpretation of coefficients

• If both Y and X are measured in standardized
  form, and

Then the bs are called standardized
coefficients. They indicate the number of
standard deviations Y will change when X changes
by one standard deviation
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BETA Coefficients Example
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Comparison of coefficients

• In general, we should NOT compare coefficients
  unless they are measured in the same units (e.g.
  dollars or inches)
• Two unit free measures are sometimes used to
  compare coefficients
• elasticities (percentage changes)
• standardized coefficients (Stand. Dev. Changes)