statistics economics homework

Econometrics 4400 HW 1

Drew Barker, adapted from Peter Nencka

Due Tuesday 9/25 at 12:45p.

You are welcome to work in teams, but please: (1) turn in your own write up and computer code, (2)

acknowledge with whom you worked at the top of your assignment. For questions using R, please print out your

code and attach it to your answers. Please submit a physical copy in class unless extenuating circumstances

prevent your attendance. I will grant 5% extra credit to your grade for this HW if you A) turn in a

physical copy of your homework that is B) completely typed out including any equations with R

code attached. Under no circumstances will I accept late homework.

Some questions have multiple possible answers. Justify your claims for full credit.

1) Education policymakers in the US are increasingly concerned that grade school (pre-college)

classrooms have too many students. The fear is that classrooms with more students have

negative effects on achievement and behavior.1

You’ve been hired as a consultant to see if the fears are justified: do classrooms with more

students have negative effects on students?

You collect data on 10,000 4th grade classrooms (200,000 students) across a random sample of

the US population in 2012.2

You estimate the following linear regressions relating the number of students in each classroom

to individual student outcomes:

𝑇𝑒𝑠𝑡𝑆𝑐𝑜𝑟𝑒𝑖 = 𝛼 + 𝛽 ∗ 𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠𝐼𝑛𝐶𝑙𝑎𝑠𝑠𝑖 + 𝑖

𝑆𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛𝑖 = 𝛾 + 𝛿 ∗ 𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠𝐼𝑛𝐶𝑙𝑎𝑠𝑠𝑖 + 𝜃𝑖

Where 𝑇𝑒𝑠𝑡𝑆𝑐𝑜𝑟𝑒𝑖 is a score on a national standardized test, and ranges from 0 to 100. Higher scores are better

𝑆𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛𝑖 is equal to 1 if a student was suspended in 2012 and 0 otherwise.

𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠𝐼𝑛𝐶𝑙𝑎𝑠𝑠 is the number of classmates a student has (ranges from 10 to 50)

1a) For each regression, what is the treatment? What is the outcome variable?

1b) In words, carefully interpret what the coefficients 𝛽 and 𝛿 mean. [Note: when a dummy

variable is the outcome, the coefficient is interpreted as the increase in the probability that the

outcome occurred.]

1 This is probably a problem in college classes too! 2 Clarification: you randomly sampled all the schools, so you don’t have to worry that you only chose schools in rich areas, for example. You have a good mix of rich, poor, urban, rural, etc schools. But classroom size (the treatment of interest) is not randomly assigned across schools.

1c) Suppose that our hypothesis is that larger classrooms harm student achievement and

increase the rate of student suspensions. What does that imply for the signs of 𝛽 and 𝛿? (are they positive or negative?)

1d) You estimate the first regression and find the following result:

𝑇𝑒𝑠𝑡𝑆𝑐𝑜𝑟𝑒𝑖 = 95 + −0.8 ∗ 𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠𝐼𝑛𝐶𝑙𝑎𝑠𝑠𝑖

Using this equation, what is the predicted test score (i.e. fitted value) for students in 20

person classes? 25 person classes? 50 person classes? What about a 0 person class? Is that last

number meaningful?

1e) You present your results from 1d to a group of education policymakers. An astute

economist asks if you are controlling for family income in your analysis. Why would it be

important to control for family income? Explain carefully the context of omitted variable bias.

1f) You listen to the economist, and estimate the following equation.

𝑇𝑒𝑠𝑡𝑆𝑐𝑜𝑟𝑒𝑖 = 𝛼 + 𝛽 ∗ 𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠𝐼𝑛𝐶𝑙𝑎𝑠𝑠𝑖 + 𝜌 ∗ 𝐼𝑛𝑐𝑜𝑚𝑒𝑖 + 𝑖

where 𝐼𝑛𝑐𝑜𝑚𝑒𝑖 is family income, and 𝜌 is the coefficient on income.

What sign do you expect 𝜌 to have? (positive or negative) Why?

1g) Using your answer in 1f, and assuming that family income is negatively correlated with class

size, how do you expect your estimate of 𝛽 to differ from the one in we got (-0.8) in 1d? Explain

in the context of omitted variable bias.

2) Read the first 5 paragraphs (more if you’re interested) of the following article in the June 1st,

2017 edition of The New York Times:

“Popular People Live Longer” https://nyti.ms/2sticEQ

Consider the following paragraph:

The results revealed that being unpopular — feeling isolated, disconnected, lonely —

predicts our life span. More surprising is just how powerful this effect can be. Dr. Holt-

Lunstad found that people who had larger networks of friends had a 50 percent

increased chance of survival by the end of the study they were in. And those who had

good-quality relationships had a 91 percent higher survival rate. This suggests that

being unpopular increases our chance of death more strongly than obesity, physical

inactivity or binge drinking. In fact, the only comparable health hazard is smoking.

2a) What’s the treatment? What’s the outcome?

2b) Assume that bolded results were generated from simple regressions without controls.3

Write down the two regressions that you would use to generate these results, carefully defining

your notation.

2c) Do the results quoted in the study seem plausible? Why or why not? What would you try to

control for? How do you think those controls would affect the bolded results?

2d) One potential problem with studies like this: even if you control for lots of variables, there

are just some things (“likeability”) that are impossible to measure. How might that affect the

results?

3) In this question you’re going to explore one of the most studied questions in empirical

economics: what is the causal effect of education on wages? The regression equation is:

𝐿𝑁𝑤𝑎𝑔𝑒𝑠𝑖 = 𝛼 + 𝛽 ∗ 𝑌𝑒𝑎𝑟𝑠𝑂𝑓𝐸𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛𝑖 + 𝑒𝑖

Where LNwages are the log wages for person i and 𝑌𝑒𝑎𝑟𝑠𝑂𝑓𝐸𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛𝑖 is person i’s number

of years of schooling. Education is the treatment, log wages are the outcome.

3a) Download “school_test.Rda” from Carmen. Load the dataset into R by going to File->Open

file, select the file you downloaded onto your computer, and select Yes when R asks if you want

to load it into the global environment.

This file contains information on 1,999 randomly sampled 28-year old men and women. The file

contains the following variables:

 id: a unique id number

 male: 0 = female, 1 = male

 log_real_wage: the natural log of real wages at age 28

 math_reading_test_score: a standardized test score that respondents took at age 14.

 years_of_experience: years of work experience

 years_of_education: years of schooling

 tenure_at_current_job: years at current job

 married: 1 = yes, 0 = no

3b) Using R, estimate the bivariate regression above relating years of education to log wages.

Report the coefficient and interpret what the coefficient on education represents. (Note: when

the outcome variable is logged, 𝛽 represents a percentage change in the outcome for each unit

increase in the treatment).

3 This doesn’t appear to be the case for some of the studies quoted here, but you would be surprised….

3b) Omitted variable bias caused by excluding student academic ability from an education

regression is often uncreatively called “ability bias.” Recall that a variable causes omitted variable

bias if it is (1) correlated with the treatment, and (2) also independently affects the outcome.

Using both parts of this definition, explain intuitively why excluding student ability from the

regression in 1a may cause omitted variable bias in our example.

3c) Add the math and reading test score to your regression from 3a. Interpret the coefficient

on the math and reading test score. Calculate the omitted variable bias caused by excluding test

scores using the coefficient on this regression and the regression in 3a.