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Understanding Dummy Variables and Interactions in Regression Analysis

Explore how dummy variables and interactions impact regression analysis, with examples and explanations on testing models. Learn about interacting dichotomous and continuous variables to enhance data analysis.

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Understanding Dummy Variables and Interactions in Regression Analysis

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  1. Interaction Terms and dummy variables in Regression

  2. Dummy variables • Dichotomous independent variables • Takes value of 0 or 1 • Gender = female (yes or no); Democrat (yes or no); South (yes or no); Klingon v Earthling; etc.

  3. Interactions between variables • Effect of one X variable may depend another X variable • Effect of X1 conditional on X2 • Effect of education on income may depend on gender (dummy variable) or age.

  4. Interactions • Tested with OLS regression • Easiest to understand when a dichotomous (dummy, categorical) variable interacted with a interval variable • Also works with continuous * continuous

  5. First, a dummy variable example • Data used is fake and includes the following variables as follows: • Race (Klingon = 0, Earthling = 1) • Education (4 <–> 16 years) • Age (25 <–> 60) • Income (100 <–> 280 dollars) • Income is Dependent variable

  6. Initial Model No interaction here: Each X variable is estimated to have its own independent association with Y (income)

  7. Initial Model Recall, how is t statistic calculated? How do we know if slope (Coef.) is significantly different than 0?

  8. Earthling 23+52.8 units Income Klingon 23 units Education -> In this case, the “slope” of X1 (Klingon =0 Earthling = 1) is an intercept difference. Slope of the effect of X (Educ) on Y (income) same for both...

  9. Earthling 23+52.8 units Income Klingon 23 units Education -> Klingon: Y = a + bX1 (12.8 * Educ) + bX2 (52.8 * 0) Earth: Y = a + bX1 (12.8 * Educ) + bX2 (52.8 * 1) Slope of the effect of X (Educ) on Y (income) same for both...

  10. Interactive Model • Does education affect income differently by race? • Find out by multiplying observations for Education by observations by race • Educationi * Earthlingi

  11. Interaction = product term

  12. Klingon Income Earthling Education -> In this case, the slope of X (Educ) on Y (income) is different for each group. It is conditional on whether one is from Earth of Klingon

  13. Specification • DO NOT omitting variables that are part of the interaction • All variables that are part of the interaction stay in the equation • e.g., don’t drop the Education and Earthling variables while leaving in Education * Earthling

  14. F-test • Omitting variables. • Not performing an F-test • Need to know if interaction contributes to model

  15. The F-test formula is where k denotes the number of variables, subscript 1 refers to original model and subscript 2 refers to the expanded model. F-Test Formula

  16. F-Test = (.74-.70)/(3-2) (1-.74)/(100-3-1) = 14.8 Critical value for F < 3.84 14.8 > 3.84 so interactive model is statistically significant

  17. Evaluating the Overall Model • Interactive terms lessen parsimony, increase difficulty of interpretation. • Don’t do unless the interactive adds explanatory power. • For OLS perform an F-test.

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