Econ 251 Fall 2015 Problem Set #2 | Complete Solution

Econ 251 Fall 2015
Problem Set #2

Part I: Data analysis (10 points in total)
This part of the problem set introduces you to using STATA for simple data analysis.

Instructions:
Following each question, please handwrite or type your answers and copy/paste the STATA output (please, use the ‘copy as picture’ option).
Download the STATA file WAGE2.dta from the CANVAS website. FYI: this file was used in a publication by Blackburn and Neumark (1992).
WAGE2.dta contains information on monthly earnings, employment history, education, demographic characteristics, and two test scores for 935 men in year 1980. In particular, it contains the following variables:
wage monthly earnings (in 1976 USD)
hours
average weekly hours of work
IQ
IQ (intelligence quotient) score
educ
years of education
age
age in years
married
=1 if the person is married
black
=1 if the person is black

1. (i) Find the average years of education for everyone in the sample (variable educ).
(ii) What are the lowest (minimum) and the highest (maximum) years of education in the sample?
(1 point)

Hint: Use the STATA command sum var1
2. (i) How many black men are there in the sample?
(ii) How many non-black men are there in the sample?
(iii) What is the percentage of non-black men in the sample?
(2 points)
Hint: Use the STATA command tab var1
3. (i) Find the average monthly wage for all men in the sample.
(ii) Find the sample mean monthly wage for blacks and non-blacks separately. Do black men in the sample earn more or less than non-black men, on average?
Hint: Use the STATA command tab var1, sum(var2)
2
In our particular case, var1 is black and var2 is wage; hence, what you would need to
type in the command window in STATA is: tab black, sum (wage)
(iii) Now find the sample average years of education for blacks and non-blacks
separately. Which group has a higher sample mean education?
(2 points)
Hint: Use the STATA command tab var1, sum(var2) again, but this time var2 is educ.
4. (i) What is the sample correlation between variables IQ (IQ test score) and educ
(years of education)? What is the sign of this correlation and what does it mean?
(2 points)
Hint: Use the STATA command corr var1 var2
(ii) Compute the average IQ test score for each year of education in the sample.
Do you find any relationship between the two variables? Does this make sense to
you?
(3 points)
Hint: Use the STATA command tab var1, sum(var2)
In this case, var1 is educ and var2 is IQ.
Part II: Statistical theory (12 points in total)
You would like to know the average wage of all working women and men between the
ages of 18 and 54 in the Ann Arbor area. You find it too costly in terms of time and
money to ask everyone about their earnings, so you randomly select 500 people from this
group, and ask them about their wage.
(i) What are the population and the population parameter of interest in this
example? What is the sample, and what the sample size, N?
(3 points)
Now, comment on each of the proposed estimators of the population mean wage, μ :
(ii) using the sample mean as an estimator of the population mean
 Is it unbiased? Show formally.
 Is it efficient? Why or why not? Show that its variance equals
Hint: we did this in class.
(3 points)
(iii) another estimator using only three observations from your sample and calculated
as: ( )
3
1
1 2 3 X  X X
Is it unbiased? Show formally.
Is it efficient? Why or why not? Explain intuitively.

OPTIONAL: Show formally that its variance equals
Hint: you solved a very similar example in Section 3.
(3 points)

(iv) another estimator using only the first and the last observation from your sample and calculated as:
Show that this estimator is biased, i.e. show that
If it is biased, can it be efficient?
Hint: see the official solutions of Section 3. (3 points)