Econometrics I Lecture 7: Dummy Variables

Transcription

1 Econometrics I Lecture 7: Dummy Variables Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology Fall / 27

2 Introduction Dummy variable: d i is a dummy variable if it only takes values of 0 and 1. Useful for studying effect of qualitative variables / event / treatment / etc. Examples: wage discrimination against women: f emale a dummy that takes 1 if the person is female and 0 otherwise. effect of clean drinking water: clean a dummy that shows whether the individuals have access to clean water. 2 / 27

3 Outline Single dummy as an explanatory variable Dummy for multiple categories Interaction terms with dummies Linear probability model Reference: Wooldridge (2013), Ch 7. 3 / 27

4 Outline Single dummy as an explanatory variable Dummy for multiple categories Interaction terms with dummies Linear probability model 4 / 27

5 Single dummy as a regressor Consider the wage equation wage = β 0 + β 1 female + β 2 educ + u β 1 : the wage gain if the person is a woman rather than a man holding education constant. Easy to show that β 1 = E [wage female, educ] E [wage male, educ] level of education is the same across expectations. Effectively, β 1 delivers a different intercept for females 5 / 27

6 Intuition: shifting intercept 6 / 27

7 Interpretation of dummies Interpretation of dummies is with reference to the omitted group In wage = β 0 + β 1 female + β 2 educ + u we dropped the male category, and β 1 was the change in wage when female relative to being male (holding education constant). Must omit one category or the constant wage = γ 0 male + γ 1 female + β 2 educ + u β 2 is the same but γ 1 = β 0 + β 1 and γ 0 = β 0. 7 / 27

8 Difference in means If we are interested in wage discrimination, one could calculate w female w male. How is this related to ˆβ 1? Notice ˆδ 1 = w female w male in wage = δ 0 + δ 1 female + u When we add more regressors we are trying to achieve the causal interpretation. The wage differential between men and women could be due to omitted characteristics that matter for wage. Women might be on average more educated. Women might on average have less experience. 8 / 27

9 Example - wage discrimination Dep. Var. wage wage wage lwage (1) (2) (3) (4) female (0.303) (0.279) (0.265) (0.0372) educ (0.0504) (0.0493) ( ) exper (0.0116) ( ) tenure (0.0212) ( ) Constant (0.210) (0.673) (0.725) (0.102) N R Mean y / 27

10 Outline Single dummy as an explanatory variable Dummy for multiple categories Interaction terms with dummies Linear probability model 10 / 27

11 Categorical variables Often we want to include a categorical variable as a control in a regression. Example: marital status (=0 single, =1 married, =2 divorced, =3 widow), city of residence,... Including a categorical variable in a regression makes no sense, instead, you must introduce a set of dummies for each category Example: Correct specification wage = β 0 +β 1 married+β 2 divorced+β 3 widow+β 4 educ+u incorrect specification wage = β 0 + β 1marital status + β 4educ + u here we impose: β 2 = 2β 1 and β 3 = 3β / 27

12 Ordinal variables For ordinal variables only the ordering matters. Example: credit rating (CR= 0, 1, 2, 3, 4) 1. Including CR in a regression makes little sense 2. Including dummies for each value of CR makes a lot of sense this is also a more flexible specification, in other words (1) is a special case of (2)! 12 / 27

13 Fixed effects Think about a dataset with n individuals in J cities. We want to model returns to education ln wage = β 0 + β 1 educ + u We might think cities might have special features that change returns to education this could lead to omitted variable bias because those living in cities with a high returns to education, would achieve higher levels of education! Is there a way to correct for this? city fixed effects: include J 1 dummies that show whether the individual lives in each city J 1 ln wage = β 0 + β 1 educ + δ j d j + u j=1 sometimes we simplify the notation and write ln wage ij = β 0 + β 1 educ ij + δ j + u ij 13 / 27

14 Fixed effects What does ln wage ij = β 0 + β 1 educ ij + δ j + u ij do? each city shifts the intercept of the regression controls for any observed and unobserved city specific characteristics that matter for wage estimation of β 1 solely relies on within city variation in wage and education effectively, ˆβ 1 is the average of estimated returns to education from the J separate city regressions. 14 / 27

15 Outline Single dummy as an explanatory variable Dummy for multiple categories Interaction terms with dummies Linear probability model 15 / 27

16 Interaction terms with dummy variables Consider wage = β 0 + β 1 female + β 2 educ + β 3 female educ + u the single f emale term shifts intercept the interaction term shifts slope How do we test for whether returns to education is the same across genders? How do we test for whether being a female has no effect on the wage? 16 / 27

17 Graphical illustration 17 / 27

18 Testing for differences in regression functions across groups Could use the same interactions idea to test whether coefficients are different for various groups. For example wage = β 0 + β 1 female + β 2 educ + β 3 female educ + β 4 exper + β 5 female exper + u Testing for β 1 = β 3 = β 5 = 0 is equivalent to testing that the same regression equation applies to men and women. 18 / 27

19 Outline Single dummy as an explanatory variable Dummy for multiple categories Interaction terms with dummies Linear probability model 19 / 27

20 Linear probability model (LPM) Consider y = β 0 + β 1 x β K x K + u where y is a dummy variable Conditional expectation is now the conditional probability of y = 1 E(y x) = 1 Pr(y = 1 x) + 0 Pr(y = 0 x) = Pr(y = 1 x) = β 0 + β 1 x β K x K Increasing x 1 by 1 unit and holding x 2,..., x K fixed, probability of y = 1 increases by β 1 units 20 / 27

21 Pros and cons of LPM advantages: easy estimation and interpretation often works well in practice disadvantages predicted probabilities could be negative or greater than 1! model gives heteroskedastic errors V ar(y x) = p(x) [1 p(x)] Logit transformation could solve the issue of unreasonable predicted probabilities Pr(y = 1 x) = eβ 0+β 1 x 1 + e β 0+β 1 x [0, 1] 21 / 27

22 Example- Female labor force participation What are the determinants of being employed? husbands income, education, experience, age, number of kids,... LPM: inlf = β 0 + β 1 nwifeinc + β 2 educ + β 3 exper + β 4 age + β 5 kids +... use MROZ.dta to estimate this model units of obs: married women 22 / 27

23 Estimation results Dep. Var. inlf (N=753) (1) (2) nwifeinc (0.0015) (0.0014) educ (0.0074) exper (0.0057) expersq (0.0002) age (0.0025) kidslt (0.0335) kidsge (0.0132) Constant (0.0359) (0.1542) R / 27

24 Predicted probabilities - LPM 24 / 27

25 Predicted probabilities - Logit 25 / 27

26 Estimation results - heteroskedasticity adjusted Dep. Var. inlf (N=753) Unadjusted Adjusted (1) (2) (3) (4) nwifeinc (0.0015) (0.0014) (0.0015) (0.0015) educ (0.0074) (0.0073) exper (0.0057) (0.0058) expersq (0.0002) (0.0002) age (0.0025) (0.0024) kidslt (0.0335) (0.0318) kidsge (0.0132) (0.0135) Constant (0.0359) (0.1542) (0.0355) (0.1523) R / 27

27 Summary In this lecture we discussed use of dummy variables as explanatory and dependent variables discussed interpretation of dummies fixed effect and interaction terms Linear Probability Model 27 / 27