micro econometrics problem set economics statistic
1. (18 points) My co-authors and I are investigating how responsive people are to prices when choosing whether or not to purchase health insurance. Our dependent variable, the take-up rate, is the fraction of people enrolled in marketplace insurance who were eligible for subsidies to purchase it. In 2014, some people with incomes between 138 percent and 400 percent of the federal poverty line (FPL) became eligible for tax credits to purchase health insurance through the marketplaces. For those eligible for subsidies, the net price of the reference insurance plan is capped at a small, but increasing share of their income. Maximum family contributions towards the price of insurance ranged from 2 percent of income for families at 138 percent of FPL to 9.5 percent for families between 300 and 400 percent of FPL.
States enacted laws that affected marketplace enrollment through a variety of mechanisms, including restrictions on navigators1 and by allowing people to remain in their nongroup plan from the prior year rather than joining a marketplace plan (grandmothering). State lawmaker’s decisions about whether to expand Medicaid eligibility2 could also affect marketplace enrollment because people who apply for but are ineligible to receive Medicaid are more likely to learn about their eligibility for subsidies to purchase Marketplace plans. Roughly half the states expanded Medicaid.
We aggregate across individuals up to a Public-Use Microdata Area (PUMA), which is similar to a county, includes at least 100,000 people, and does not overlap or span multiple states. For each PUMA there are 16 observations broken out by gender(male or female), age category(aged 25-34, 35-44, 45-54, or 55-64), and the ratio of income to the federal poverty line category(138-250 percent of FPL or 250-400 percent of FPL). For each year, there are roughly 1,500 PUMAs so 16 × 1, 500 = 24, 000 observations. Our data set includes the years 2015-2017.
yit =Take-up rate of marketplace plans for PUMA, gender, age, and income group i in year t
pit =Average net price for the reference insurance plan for group i in year t
x1i =Vector of gender and age dummies for 7 groups excluding male aged 20-34 years old
x2i =Vector of income to FPL and Medicaid eligibility expansion dummies for 3 groups
=excluding income 250-400 percent of FPL in a state that did not expand Medicaid
x3it=Vector of demographic, market, and policy variables for group i in year t
x3 includes the uninsured rate in the state in 2013, number of insurers participating in the PUMA in year t, mean age in PUMA, gender, age, income group i in year t, and mean income in PUMA, gender, age, income group i in year t.
You want to learn the effect of net price on take-up of insurance, α.
yit = β0 + α pit + β1x1it + β2x2it + β3x3it + γi + δt + it
1
Navigators are individuals or organizations trained and able to help people as they look for health coverage options through the marketplace, including completing eligibility and enrollment forms. 2Medicaid is public insurance offered to low income people, and the eligibility expansions were for adults below 138 percent of poverty.
(a) 4 points If you estimated this model using generalized least squares with 2015 and 2016 dummies, what implicit assumption(s) are you making about γi and it to justify your estimator for α? (b) 4 points Do you think that assumption is justified?
(c) 6 points What is the reason for including state fixed effects, if any? Under what conditions will your estimator ˆαF E be a good estimator for the effect of net price on take-up of insurance? Be sure to clarify which properties make your estimator good. Do you believe that these conditions hold?
(d) 4 points What are the advantages and disadvantages of having PUMA fixed effects rather than state fixed effects in your model specification?
2. (45 points) The respiratory disease named “coronavirus disease 2019†(abbreviated “COVID19â€) started spreading in the United States in 2020. It is spread more rapidly with more person-to-person contact. There are laboratory tests that can identify the virus that causes COVID-19, but their availability is limited and set by the federal government. Death rates have tended to be higher for people aged 65 or older and for people with high risk conditions – lung disease, moderate or severe asthma, and serious heart conditions. A mechanical ventilator is a machine that’s used to support patients with severe respiratory conditions that impact the lungs, including pneumonia. Availability of Intensive Care Unit (ICU) beds can decrease the death rate from COVID-19 because they are the hospital units typically equipped to treat patients with respiratory problems that require ventilators. The Society of Critical Care Medicine found that, in the United States, the vast majority of ICU beds are in large metropolitan areas and only 1 percent are in rural areas.
Suppose that you have a data set with the following information at the county and state level: Deathij =ratio of deaths attributable to confirmed cases of COVID-19 in county i in state j
Elderlyij =share of population aged 65 or older in county i in state j
RiskyCondij =share of population with high risk conditions in county i in state j
V entilatorsij=ventilators available per thousand people in county i in state j
ICUij =intensive care unit beds available per thousand people in county i in state j
Ruralij =share of population living in a rural area in county i in state j
T estsij =number of COVID-19 tests that are available per thousand people in county i in state j
T ravelij =share of population that travelled internationally last year living in county i in state j
You are interested in learning the extent to which the share of the population that is elderly (aged 65 or older) affects the rate of death from COVID-19. You run an ordinary least squares regression to estimate the βs in:
Deathij = β0 + β1 Elderlyij + β2RiskyCondij + β3V entilatorsij + β4ICUij + ij
(a) 6 points Explain why omitting Rural from a regression of Death (rate) on Elderly, RiskyCond, Ventilators, and ICU would or would not bias the coefficient estimate on Elderly, βˆ 1.
(b) 6 points If you did include Rural and were willing to assume that the Multiple Linear Regression model applied, would you use the default standard errors for your measure of precision of βˆ 1? If not, why not and how would you calculate the appropriate standard error?
(c) 6 points Government-mandated social distancing (school closures, non-essential service closures, and shelter-in-place orders) can dramatically influence the course of the pandemic and decrease fatality rates by slowing the need for medical services. Let SDj be a measure of the extent to which the governor mandates social distancing in state j. If you were to include SD in the ordinary least squares regression, would the coefficient on SD be an unbiased and/or consistent measure of the effect of the extent of a social distancing mandate on the death rate from COVID-19?
(d) 6 points What are the two necessary assumptions to make an instrumental variable, z, valid?
(e) 2 points Assuming you had a valid IV for mandated social distancing, SD, what nice property does your estimator have?
(f) 9 points Are the following candidate variables valid instruments? Why or why not?
i. Indicator equal to 1 if the (state) governor is the same political party as the president = z1. The president vocally opposed early imposition of mandating social distancing.
ii. COVID-19 tests available in the state, T estsj = z2
iii. Share of state population that travelled internationally last year, T ravelj = z3.
(g) 6 points Choosing any or all of the above IVs, how would you estimate the extent to which the share of the population that is elderly affects the rate of death from COVID19? Be specific about your model specification and estimator.
(h) 4 points Suppose you believed that more than one of your Instrumental Variables was potentially valid. What test would you use to check the validity of your IVs?
3. (27 points) You are interested in the effect of civic engagement (community service work when in high school) on the probability of graduating from college. In theory, civically active students might gain a greater understanding of the value of education and thus stay in school longer.
P assi=1 if ith person graduated from college, 0 otherwise
Ci =1 if ith High School student performed community service in 1992, 0 otherwise
Bi =1 if ith High School student identifies as black, 0 otherwise
Fi =1 if ith High School student identifies as female, 0 otherwise
Si =1 if ith High School student lived in a single parent household, 0 otherwise
Pi =Highest education level attained by a parent of ith High School student
Ii =Income of parents of ith High School student (in thousands of dollars in 1988)
(a) 6 points Under what conditions will your estimator b1,OLS be a good estimator for β1 (the effect of civic engagement in high school on the probability of college graduation)? Be sure to clarify which properties make your estimator good. Do you believe that these conditions hold?
(b) 4 points What are the disadvantages, if any, of the ordinary least squares estimator for the effect of civic engagement (community service work when in high school) on the probability of graduating from College? You do not need to reiterate your concerns from part a.
(c) 5 points Suppose you plan to use Maximum Likelihood Estimator for the effect of interest. Write down the (conditional) Log Likelihood function for the Logit model.
(d) 4 points Suppose that some schools required community service and you could distinguish between students who engaged in community service that was required from those who voluntarily engaged in community service. How would that help you to estimate β1?
(e) 5 points Test, at the 5% significance level, the null hypothesis that civic engagment does not affect the probability of graduating from college, all else equal. Be sure to state your alternative hypothesis, critical value, and write your conclusion in a sentence. The Logit model estimated with 9419 observations, yielded the following coefficient estimates and standard errors. explanatory Logit variable βˆ std err C=CivicInd 0.276 0.019 B=BlackInd -.152 0.029 F=FemaleInd 0.142 0.014 S=SinglePInd 0.100 0.125 P=Parental Education 0.404 0.033 I=Income 0.004 0.00003 constant -1.18 .05
(f) 3 points Can you tell whether the partial effect of civic engagement on the probability of passing is positive or negative? Or would you need other information for that (and what information)?
