Section 5: Dummy Variables and Interactions

Transcription

1 Section 5: Dummy Variables and Interactions Carlos M. Carvalho The University of Texas at Austin McCombs School of Business 1

2 Example: Detecting Sex Discrimination Imagine you are a trial lawyer and you want to file a suit against a company for salary discrimination... you gather the following data... Gender Salary 1 Male Female Female Female Male Female

3 Detecting Sex Discrimination You want to relate salary(y ) to gender(x )... how can we do that? Gender is an example of a categorical variable. The variable gender separates our data into 2 groups or categories. The question we want to answer is: how is your salary related to which group you belong to... Could we think about additional examples of categories potentially associated with salary? MBA education vs. not legal vs. illegal immigrant quarterback vs wide receiver 3

4 Detecting Sex Discrimination We can use regression to answer these question but we need to recode the categorical variable into a dummy variable Gender Salary Sex 1 Male Female Female Female Male Female Note: In Excel you can create the dummy variable using the formula: =IF(Gender= Male,1,0) 4

5 Detecting Sex Discrimination Now you can present the following model in court: How do you interpret β 1? Salary i = β 0 + β 1 Sex i + ɛ i E[Salary Sex = 0] = β 0 E[Salary Sex = 1] = β 0 + β 1 β 1 is the male/female difference 5

6 Detecting Sex Discrimination SUMMARY OUTPUT Salary i = β 0 + β 1 Sex i + ɛ i Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 208 ANOVA df SS MS F Significance F Regression E-07 Residual Total Coefficientstandard Erro t Stat P-value Lower 95% Upper 95% Intercept E Gender E ˆβ 1 = b 1 = on average, a male makes approximately $8,300 more than a female in this firm. How should the plaintiff s lawyer use the confidence interval in his presentation? 6

7 Detecting Sex Discrimination How can the defense attorney try to counteract the plaintiff s argument? Perhaps, the observed difference in salaries is related to other variables in the background and NOT to policy discrimination... Obviously, there are many other factors which we can legitimately use in determining salaries: education job productivity experience How can we use regression to incorporate additional information? 7

8 Detecting Sex Discrimination Let s add a measure of experience... Salary i = β 0 + β 1 Sex i + β 2 Exp i + ɛ i What does that mean? E[Salary Sex = 0, Exp] = β 0 + β 2 Exp E[Salary Sex = 1, Exp] = (β 0 + β 1 ) + β 2 Exp 8

9 Detecting Sex Discrimination Exp Gender Salary Sex 1 3 Male Female Female Female Male Female

10 Detecting Sex Discrimination Salary i = β 0 + β 1 Sex i + β 2 Exp + ɛ i Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 208 ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Sex Exp Salary i = Sex i Exp i + ɛ i Is this good or bad news for the defense? 10

11 Detecting Sex Discrimination Salary i = { Expi + ɛ i females Exp i + ɛ i males Experience Salary 11

12 More than Two Categories We can use dummy variables in situations in which there are more than two categories. Dummy variables are needed for each category except one, designated as the base category. Why? Remember that the numerical value of each category has no quantitative meaning! 12

13 Example: House Prices We want to evaluate the difference in house prices in a couple of different neighborhoods. Nbhd SqFt Price

14 Example: House Prices Let s create the dummy variables dn1, dn2 and dn3... Nbhd SqFt Price dn1 dn2 dn

15 Example: House Prices Price i = β 0 + β 1 dn1 i + β 2 dn2 i + β 3 Size i + ɛ i E[Price dn1 = 1, Size] = β 0 + β 1 + β 3 Size (Nbhd 1) E[Price dn2 = 1, Size] = β 0 + β 2 + β 3 Size (Nbhd 2) E[Price dn1 = 0, dn2 = 0, Size] = β 0 + β 3 Size (Nbhd 3) 15

16 Example: House Prices Price = β 0 + β 1 dn1 + β 2 dn2 + β 3 Size + ɛ SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 128 ANOVA df SS MS F gnificance F Regression E-31 Residual Total Coefficients Standard Error t Stat P-value Lower 95%Upper 95% Intercept dn dn size Price = dn dn Size + ɛ 16

17 Example: House Prices Price Nbhd = 1 Nbhd = 2 Nbhd = Size 17

18 Example: House Prices SUMMARY OUTPUT Price = β 0 + β 1 Size + ɛ Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 128 ANOVA df SS MS F gnificance F Regression E-11 Residual Total Coefficientsandard Err t Stat P-value ower 95%pper 95% Intercept size Price = Size + ɛ 18

19 Example: House Prices Price Nbhd = 1 Nbhd = 2 Nbhd = 3 Just Size Size 19

20 Back to the Sex Discrimination Case Experience Salary Does it look like the effect of experience on salary is the same for males and females? 20

21 Back to the Sex Discrimination Case Could we try to expand our analysis by allowing a different slope for each group? Yes... Consider the following model: Salary i = β 0 + β 1 Exp i + β 2 Sex i + β 3 Exp i Sex i + ɛ i For Females: For Males: Salary i = β 0 + β 1 Exp i + ɛ i Salary i = (β 0 + β 2 ) + (β 1 + β 3 )Exp i + ɛ i 21

22 Sex Discrimination Case How does the data look like? Exp Gender Salary Sex Exp*Sex 1 3 Male Female Female Female Male Female

23 Sex Discrimination Case Salary = β 0 + β 1 Sex + β 2 Exp + β 3 Exp Sex + ɛ Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 208 ANOVA df SS MS F Significance F Regression E-45 Residual Total CoefficientsStandard Erro t Stat P-value Lower 95% Upper 95% Intercept Sex Exp Sex*Exp Salary = 34 4Sex Exp Exp Sex + ɛ 23

24 Sex Discrimination Case Experience Salary Is this good or bad news for the plaintiff? 24

25 Variable Interaction So, the effect of experience on salary is different for males and females... in general, when the effect of the variable X 1 onto Y depends on another variable X 2 we say that X 1 and X 2 interact with each other. We can extend this notion by the inclusion of multiplicative effects through interaction terms. Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 (X 1 X 2 ) + ε E[Y X 1, X 2 ] X 1 = β 1 + β 3 X 2 We will pick this up in our next section... 25

26 Example: College GPA and Age Consider the connection between college and MBA grades: A model to predict McCombs GPA from college GPA could be GPA MBA = β 0 + β 1 GPA Bach + ε Estimate Std.Error t value Pr(> t ) BachGPA ** For every 1 point increase in college GPA, your expected GPA at McCombs increases by about.26 points. 26

27 College GPA and Age However, this model assumes that the marginal effect of College GPA is the same for any age. It seems that how you did in college should have less effect on your MBA GPA as you get older (farther from college). We can account for this intuition with an interaction term: GPA MBA = β 0 + β 1 GPA Bach + β 2 (Age GPA Bach ) + ε Now, the college effect is E[GPAMBA GPA Bach Age] GPA Bach = β 1 + β 2 Age. Depends on Age! 27

28 College GPA and Age GPA MBA = β 0 + β 1 GPA Bach + β 2 (Age GPA Bach ) + ε Here, we have the interaction term but do not the main effect of age... what are we assuming? Estimate Std.Error t value Pr(> t ) BachGPA e-05 *** BachGPA:Age ** 28

29 College GPA and Age Without the interaction term Marginal effect of College GPA is b 1 = With the interaction term: Marginal effect is b 1 + b 2 Age = Age. Age Marginal Effect