Economics 420 (sections 2 and 3) Fall Semester 2015 Problem Set #4

Directions: Following each question, type or handwrite your answers and copy/paste the Stata
output (use the ‘copy as picture’ option). Staple all pages together — assignments turned in
unstapled will be returned with a grade of zero. (Only stapling is acceptable — paper clips and
other methods of binding are not acceptable.) Also, if we cannot discern the meaning of your
work, your response will be assumed wrong.
This problem set introduces you to regression with dummy variables and interaction terms, and
hypotheses tests involving more than one parameter. It uses Stata file BEAUTY.dta, which
contains the following variables used by Hamermesh and Biddle (American Economic Review
1994):
wage hourly wage
exper years of workforce experience
looks physical attractiveness score ranging from 1 to 5
black =1 if black
female =1 if female
educ years of schooling

Problem 1 (5 points total)
1.1. (1 point) Estimate the simple linear regression model:
lwage = ?0 + ?1female + u
Interpret the OLS estimates of the intercept and the slope on female.
1.2. (3 points) Now estimate the multiple linear regression model:
lwage = ?0 + ?1female + ?2educ + u.
What are the OLS estimates of the slope on female and educ and how do you interpret them?
(You do not need to comment on the intercept.)
1.3 (1 point) Based on your estimates in part 1.2, draw a graph with educ on the X-axis and
lwage on the Y-axis showing the regression lines for females and males. Hint: We solved a similar
problem in class.

Problem 2 (6 points total)
Consider the following three population models for log-earnings:
log(wage) = ?0 + ?1educ + uf (for men only)
log(wage) = ?0 + ?1educ + um (for women only)
log(wage) = ?0 + ?1educ + ?2female + ?3educ•female + ub (for both women and men)
2.1. (2 points) Write each of the four ? coefficients in terms of ?0, ?1, ?0, and ?1. That is, show
how ?0 , ?1, ?2, and ?3 are related to ?0, ?1, ?0, and ?1. You need to write down four equations:
?0 = ..., ?1 = ..., ?2 = ..., and ?3 = .... (Notes: The first equation is ?0 = ?0. This question is asking
about the population parameters, not the OLS estimates. We did something similar to this in
class.)
2.2. (3 points) Now use dataset BEAUTY.dta to estimate these three regressions and interpret
all of the estimated parameters (?0, ?1, ?2, and ?3) of the third model. Which of these
coefficients are statistically significant at a significance level of 0.01?
Hints: You will need to generate the log-wage variable. To estimate the first two models, type in
Stata:
reg lwage educ if female==0
reg lwage educ if male==1
To estimate the third model, first generate a variable equal to the product of variables educ and
female (the interaction term) by typing:
gen educ•female = educ*female
Then estimate the regression:
reg lwage educ educ•female
2.3 (1 point) Based on your estimates in 2.2, draw a graph with educ on the X-axis and lwage on
the Y-axis showing the regression lines for males and females.
Problem 3 (9 points in total)
The variable looks from dataset BEAUTY.dta contains each persons’s score on their physical
attractiveness, as ranked by an interviewer. Attractiveness was coded in five categories:
1=homely, 2=quite plain, 3=average, 4=good looking, and 5=strikingly beautiful/handsome.
3.1. Create three dummy variables that represent a person’s looks as follows: the first variable
(belowaverage) equals 1 if looks is less than 3, 0 otherwise; the second (average) equals 1 if looks =
3, 0 otherwise; and the third (aboveaverage) equals 1 if looks is greater than 3, 0 otherwise. Note:
no submission for this part is required.
Hint: All you need to do for this part is type in Stata:
gen belowaverage = (looks<3)
gen average = (looks==3)
gen aboveaverage =(looks>3)
3.2. (3 points) Now estimate the following model for log-earnings (lwage) for women:
lwage = ?0 + ?1belowaverage + ?2aboveaverage + u
Interpret the OLS estimates of ?0, ?1, and ?2. (Hint: To estimate the model for women, type in
Stata:
reg lwage belowaverage aboveaverage if female==1
Note that you cannot include all the dummies in the regression — in this case we excluded
average.)
2
Now use the regression results to test the following hypotheses. To get full credit write down
the null hypothesis, the t-statistic, the p-value, whether you reject the null, and why. Include your
Stata output.
3.3. (2 points) Test the hypothesis that women with below average looks earn the same logwage
as women with average looks. Use a significance level of 10%.
3.4. (2 points) Test the hypothesis that women with above average looks earn the same log-wage
as women with below average looks. Use a significance level of 5%.
3.5. (2 points) Test the hypothesis that how a woman looks does not affect her log-earnings.