Econometrics

ECON 4600 - Fall 2022

Problem Set 3 Due: November 28th at 3:50 pm - submit at recitation.

Instructions: Instructions: You should submit one answer per group and list all your group members’ names. You need to submit both your written answer and your R file used for programming. Email your R file to Yiyu and hand in your hard copy of written answer in class. Both have to be done BEFORE the due date and early submission is also accepted. You should type up your answers - except when you need to write down equations. Please present your answer in a professional manner. When you are asked to compute/plot a statistics/graph, please include the statistics/graph in your submission. No late submissions are accepted without prior arrangement!

1. Regression Analysis: (50 points) James wants to study how a worker’s personal income ex- plains his/her access to quality healthcare. He plans to use the simple linear regression model,

qualityi = β0 + β1incomei + εi, for i = 1, ..., N.

Both qualityi and incomei are measured in dollars. James applies the OLS estimator on sample data to obtain estimates called β̂0 and β̂1. However, James’ colleague Michael discovers flaws in James’ work. For your convenience, you may assume the table describes the sample,

Observation Quality Income State Denis $10,000 $30,000 Alabama Fred $25,000 $70,000 Florida

Vikesh $19,000 $60,000 Alaska Felicia $21,000 $65,000 Alabama Craig $12,000 $45,000 Florida

(a) James decides to use a different linear regression model,

qualityi = β0+β1incomei+β2incomeiFloridai+β3incomeiAlabamai+β4incomeiAlaskai+εi,

where

Floridai =

{ 1 if the person is in Florida

0 otherwise ,

Alaskai =

{ 1 if the person is in Alaska

0 otherwise

and

Alabamai =

{ 1 if the person is in Alabama

0 otherwise .

Write down the regressor matrix X for Denis, Fred, Vikesh, Felicia, and Craig. Does your X matrix violate the rank condition? Do you have perfect multicollinearity? Explain.

(b) James and Michael agree to meet again to discuss the simple linear regression,

qualityi = β0 + β1incomei + εi.

Michael thinks James’ estimates are not consistent because James didn’t include parental income as a regressor. James thinks Michael is nitpicking and he disagrees with Michael’s assessments. Using the OLS bias formula, describe the conditions for when James is wrong. Do you think Michael is right? Explain why or why not.

(c) Following the discussion in (d), Michael convinces James that the OLS estimate is incon- sistent because of the omission of parental income. However, James observes parental years of education but not parental income. So James proposes to use father’s year of education, fatheduci, as instruments for an IV estimator of β1, called β̂IV . Provide conditions for β̂IV to be consistent. Do you think those conditions are reasonable in this problem? Explain why or why not.

(d) Suppose James tries a different strategy. He now uses fatheduci to proxy for the father’s income, fathinci. The population model is

qualityi = β0 + β1incomei + β2fathinci + εi,

with E[εi|incomei, fathinci] = 0. But James estimates the model,

qualityi = β0 + β1incomei + β2fatheduci + ε̃i.

James acknowledges fatheduci measures fathinci imperfectly, i.e.

fatheduci = fathinci + ui.

It turns out that β̂2 is statistically insignificant. Now James confidently asserts it was a complete waste of time to worry about fathinci as a control. Provide a critique of James’ assertion.

2. California Test Scores: (100 points) This exercise focuses on the CATestScoreData data file. You can download both the data file and its accompanying README file from Canvas. Please read the README file to familiarize yourself with the data set.

(a) To start off the exercise, you should estimate the simple linear regression.

MATHi = β0 + β1SIZEi + εi (1)

Notation: MATHi is the school’s average math score. SIZEi is the school’s student-to- teacher ratio. INCOMEi is the district’s average income. MEALi is the percentage of students qualifying for reduced-price lunch. ENGi is the school’s percent of English learners.

� Use the “stargazer” command to tabulate your findings.

� Interpret the meaning of β̂1.

� Some educators believe the student’s success is largely determined by the student-to- teacher ratio. What percentage of the math score performance can be explained by the student-to-teacher ratio alone?

(b) Now consider the multiple linear regression model,

MATHi = β0 + β1SIZEi + β2INCOMEi + εi (2)

� Calculate the correlation coefficient between SIZEi and INCOMEi.

� Use the stargazer command to tabulate the two regressions, (1) and (2).

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� Is β̂1 statistically significant in (2)? In addition, you should report its confidence inter- val.

� Use the omitted variable bias formula to explain why β̂1 changed from (1) to (2).

� How does your finding add to the socioeconomic vs. class size question on the student’s outcome?

(c) Omitted variable bias could have driven your conclusion in (b). To assuage that concern, you should consider the multiple linear regression model

MATHi = β0 + β1SIZEi + β2INCOMEi + β3MEALi + β4ENGi + εi (3)

� Use the stargazer command to tabulate the three regressions, (1), (2), and (3).

� Is β̂1 statistically significant in (3)?

� What does regression (3) add to the socioeconomic vs class size question on the student’s outcome?

(d) Is your analysis in (c) driven by outliers?

� Conduct the leave-one-out analysis and remove outliers by the two standard deviation rule.

� Use the stargazer command to tabulate the original regression (3) and the regression (3) without outliers.

� Report your findings.

(e) Is your analysis in (b) driven by heteroskedasticity?

� Take a two-pronged approach to assess the homoskedastic assumption.

� First, you should produce residual-squared plots for each four regressors.

� Second, you should use the BGP regression:

V ar(εi|Xi) = γ0 + γ1SIZEi + γ2INCOMEi + γ2MEALi + γ4ENGi + ui,

to test for heteroskedasticity at the 5% false positive rate.

� If heteroskedasticity matters, does it change your analysis? Use the stargazer command to produce the output for comparison.

(f) Now you should find the “best” predictive model for MATHi.

� Suggest at least ten candidate regressors to use for your model.

� When selecting regressors, you should take account of nonlinearities and mean indepen- dence.

� You should defend your choice of variables.

� Now you should use a model selection method to find your final model.

� Detail your steps.

� Remove outliers - like in d).

� Suggest an appropriate BGP regression to test for heteroskedasticity. Defend your choice.

� Produce a 95% prediction interval for the average math score of a school having these characteristics:

– average reading score is 656,

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– total number of students enrolled is 3000,

– computers per student is 0.2,

– the district average income is $40,000,

– 45% of students qualify for reduced-price lunch,

– the student-teacher ratio is 20,

– $6000 expenditures per student,

– percent of English learners is 20%, and

– 20 percent of students qualify for Calworks.

Justify your approach.

3. Linear Probability Model: (100 points) This exercise makes you use the NLS Public Investi- gator - the link is here. Pick the National Longitudinal Survey of Youth 1979.

(a) Download the data set having the listed variables.

� Martial status in 2018. (Variable: T8219300)

� Total net family income recorded in 2018. (Variable: T8218700)

� Whether individual lived in a rural/urban area in 2018. (Variable: T8221200)

� Condition of physical health in 2018. (Variable: T7981600)

� Family size in 2018. (Variable: T8218600)

(b) Clean the data.

� Keep either married (=1), divorced(=3), or separated (=2) observations.

� Keep observations with positive net family income (>0).

� Keep observations who are known to be either in a rural/urban area (0 or 1).

� Keep observations who have known health status (>0).

� Keep observations having a positive number of family size (>0).

(c) Construct these variables.

� Outcome variable:

maritiali =

{ 1 if married

0 if separated or divorced .

� Income regressor:

incomei = T8218700

10000 .

� Urban regressor:

urbani =

{ 1 if person lives in an urban area

0 if person lives in a rural area .

� Healthy regressor:

healthyi =

{ 1 if T7981600≤ 4

0 if T7981600= 5 .

� Family size regressor: famsizei = T8218600

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(d) Use your constructed variables to run a linear probability model. Use the stargazer command to tabulate your results.

(e) Interpret your estimated coefficients of your linear probability model.

(f) The individual of interest is someone who is healthy (=1), lives in a rural area, has a family of two, and earns $2 millions per year in net family income. You should use the Bayesian Classifier to predict the person’s martial status.

(g) What is the fractions correctly predicted?

4. Panel Regression: (60 Points) (Modified Stock and Watson E10.1) Some U.S. states have enacted laws that allow citizens to carry concealed weapons. These laws are known as “shall- issue” laws because they instruct local authorities to issue a concealed weapons permit to all applicants who are citizens, are mentally competent, and have not been convicted of a felony. (Some states have some additional restrictions.) Proponents argue that if more people carry concealed weapons, crime will decline because criminals will be deterred from attacking other people. Opponents argue that crime will increase because of accidental or spontaneous use of the weapons. In this exercise, you will analyze the effect of concealed weapons laws on violent crimes. Use the “Guns.csv” file to answer the questions. The Guns Description file gives a detailed descriptions of variables. The data contains information about 50 U.S. states plus the district of Columbia for the years 1977 through 1999.

1. Is data set a panel, a cross-section, or a time series? If it is a panel, then you should describe what the entity is and what the period is.

2. Consider the linear regression model,

log(vioit) = β0 + β1shallit +X ′itβ−1 + εit,

where control variables are packed into

X ′it = (incar rateit, densityit, avgincit, popit, pb1064it, pw1064it, pm1029it)

and β−1 = ( β2 β3 β4 β5 β6 β7 β8

)′ . Estimate this regression and interpret the esti-

mated β1. Do you think it is large or small in real-world sense? Make sure to use clustered standard errors - also explain why that is relevant here.

3. Suggests a variable that varies across states but plausibly varies little-or not at all- over time and that could cause omitted variable bias in the regression.

4. Consider the fixed effect regression,

log(vioit) = β0 + β1shallit +X ′itβ−1 + αi + εit.

(a) Does this regression have a perfect multicollinearity problem? If so explain why and provide a modification to fix it.

(b) Now estimate the regression. Does the estimate β1 look more sensible or less sensible?

(c) The economy changes over time and workers’ opportunities change with it. Conse- quently, the incentives to commit crime also change with time. Does this fact pose a threat to your estimate of β1? Explain why or why not.

(d) Modify your regression to address the problem in (c). Then estimate your new regres- sion. Is your estimate of β1 now smaller or bigger?

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