# Economics 420 Fall Semester 2015 Problem Set #4

March 8, 2016

**Economics 420 Fall Semester 2015 Problem Set #4 in $16 only**

**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 (below average) 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 l wage below average above average 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 log wage 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.

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