Multiple Regression III

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Multiple Regression III
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Topic 1. Dummy Variables

• Sometimes, you have categorical variables that
  you would like to use as independent variables in
  a regression.
• Examples might include gender, religion, race,
  etc..
• For dichotomous variables, you simply include a
  variable that is coded as 1 for one category and
  zero for the other.
• e.g. female 1 and male 0.
• In this case, the regression coefficient is
  interpreted as the difference in the dependent
  variable between men and women.
• We call these kinds of variables dummy variables.

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• Note to self
• Draw a scatterplot on the board using Xs to
  denote men and Os to denote women, where men the
  distribution of men is shifted above the
  distribution of women, but the slope of the line
  is essentially the same.

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Dummy Variables cont.

• Polychotomous variables are trickier than
  dichotomous variables.
• One variable would not be very effective at
  measuring the effect of a polychotomous variable
  whose categories lack a clear ordering.
• For example, consider the regression model
• Income B0 B1 Race B2 Yrs Education B3
  Yrs Experience
• What is wrong with this model?

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Dummy Variables cont.

• If there were only two races, then we could
  estimate the model with race coded such that
  blacks 1 and whites 0.
• Income B0 B1 Race B2 Yrs Education B3 Yrs
  Experience
• However, there are multiple categories of race,
  and we wouldnt want to say that, Asians had a
  bit more race than whites but less race than
  blacks.
• Instead, we use a set of dummy variables in
  order to describe the polychotomous variable.
  Each dummy variable in the set describes whether
  or not an individual belongs to a certain
  category in the model.

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Dummy Variables cont.

• Example to describe a polychotomous variable for
  race that contained the categories black, white ,
  Asian, and Latino, we could use the following
  variables
• BlackVar 1 for blacks, 0 for all others
• WhiteVar 1 for whites, 0 for all others
• LatinoVar 1 for Latinos, 0 for all others
• Each of these variables would then be included in
  a regression model
• Income B0 B1 BlackVar B2 WhiteVar B3
  LatinoVar
• B4 Yrs Education B5 Yrs Experience
• Why dont we include a variable for Asians?
• The regression coefficients are then interpreted
  as the effect of being in Category X as compared
  to being in the excluded group. Therefore, in the
  above example, B1 gives the increase/decrease in
  income for blacks relative to Asians.

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Dummy Variables cont.

• Example from Crime and Temp Data set.

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Topic 2. Interaction Terms

• Sometimes theory predicts that the regression
  coefficient for a particular independent variable
  will not be constant for all observations.
• For example, the effect of education on income
  might be different for men and women.
• In this case, the following regression model will
  not be adequate
• Income B0 B1 Male B2 Yrs Education B3
  Yrs Experience
• Why not? What would you do instead?

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Interactions Terms cont.

• A superior method would be to relax the
  assumption that B2 was identical for both men and
  women by adding an additional variable.
• For each observation, define this new variable as
  follows maleXeduc Male Yrs Education
• This new variable allows us to measures how the
  effect of education differs for men and women
  when we estimate the following regression model.
• Income B0 B1 Gender B2 Yrs Education B3
  Yrs Experience
• B4 maleXeduc
• Suppose B4 3, then that would mean that for
  each additional year of education, a male would
  gain three additional units of income compared to
  a female.
• Continuing the example, suppose you had a male
  and a female with 12 years of school. If each
  went to college for 4 years, then the males
  income would rise 12 units of income more than
  the females.

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Interaction terms

• Draw a scatterplot to illustrate the difference
  between an intercept shift with just a dummy
  variable and a slope shift due to an interaction
  term.