Optional!!! This problem set is entirely optional. If you submit it, it will replace your lowest problem set grade (unless it is lower than your lowest grade). If you do not submit this assignment, your grade will be unchanged. Either way, I encourage you to work through the problems to prepare for the final exam.
Due Upload your answer on Canvas before 11:59PM on Wednesday, 11 March 2026.
Important You must submit your answers as an HTML or PDF file, built from an RMarkdown (.Rmd) or Quarto (.qmd) file. Do not submit the .Rmd or .qmd file. You will not receive credit for it.
If we ask you to create a figure or run a regression, then the figure or regression results should be in the document that you submit (not just the code—we want the actual figure or regression output with coefficients, standard errors, etc.).
Integrity If you are suspected of cheating, then you will receive a zero—for the assignment and possibly for the course.
README! The data for this problem set are stored in data-ps3.csv. They build on the county-level life-expectancy dataset from Problem Set 1 and add two new tax-related variables from the original county-characteristics source:
Variable names and descriptions
Variable name
Variable type
Variable description
county_name
character
County name
county_code
integer
County FIPS code
state_abb
character
State abbreviation
income_quartile
character
Income quartile (1 or 4)
life_exp
numeric
Life expectancy (years)
pct_uninsured
numeric
Percent uninsured
poverty_rate
numeric
Share below poverty line
pct_religious
numeric
Percent religious
pct_black
numeric
Percent Black
pct_hispanic
numeric
Percent Hispanic
unemployment_rate
numeric
Unemployment rate
median_hh_inc
numeric
Median household income
is_urban
integer
Urban county indicator (1 or 0)
pop
integer
County population
pop_density
numeric
Population density
pct_smoke
numeric
Percent who smoke
pct_obese
numeric
Percent obese (BMI)
pct_exercise
numeric
Percent who exercise
tax_rate
numeric
Local tax rate
tax_prog
numeric
Tax progressivity
Recall that the data came from a study titled The Association Between Income and Life Expectancy in the United States, 2001-2014 (by Raj Chetty, Augustin Bergeron, David Cutler, Benjamin Scuderi, Michael Stepner, and Nicholas Turner).
You can find more information about the life-expectancy variables here and the county characteristics here.
Objective This problem set has three goals:
Practice reasoning carefully about causality in observational data.
Implement and interpret instrumental-variables and two-stage least squares estimators in R.
Review the main ideas from lecture to prepare for the final exam.
2 Setup
[00] Load your R packages and the dataset (data-ps3.csv). You will likely want tidyverse, scales, fixest, and here.
Answer I am using read_csv() from the tidyverse package to load the data.
Rows: 3106 Columns: 20
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): county_name, state_abb
dbl (18): county_code, income_quartile, life_exp, pct_uninsured, poverty_rat...
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
3 Section 1: Causality and IV
In this section, we will study the relationship between smoking and life expectancy. A natural starting point is the regression
The sample contains 3,106 observations, 1,554 counties, and 50 state abbreviations.
The mean tax rate is 3%, and the mean tax progressivity is 0.889.
Finally, for the lowest income quartile, the mean life expectancy is 78.87 years; the mean smoking share is 28%. For the highest income quartile, the mean life expectancy is 85.97 years; the mean smoking share is 13%.
[03] Create a figure that helps you visualize the raw relationship between smoking and life expectancy. Make the figure informative. At a minimum, it should
put pct_smoke on the horizontal axis,
put life_exp on the vertical axis,
distinguish the two income quartiles somehow (e.g., make two separate plots or color by quartile),
Bonus: Include fitted regression lines in your plot.
Hint: In ggplot2, you can use geom_smooth(method = 'lm') to add a fitted regression line to a scatter plot.
Briefly describe what you see.
Answer I am using a scatter plot with separate colors for the two income quartiles.
The raw relationship is negative: counties/quartiles with higher rates of smoking tend to have lower life expectancy. The higher-income quartile also lies above the lower-income quartile, suggesting a large income-related gap in life expectancy. There are also a few outliers with very high smoking rates (100%).
[04] Estimate the following model by OLS using heteroskedasticity-robust standard errors:
where \(1\{ \text{(Income quartile)}_i = 1 \}\) is an indicator for being in the lowest income quartile (with the lowest quartile as the reference group).
Report the results (e.g., in a table).
Then interpret the coefficient on pct_smoke.
Hint: You have a few options for how to implement this regression’s indicator variable:
create a new variable in the dataset that is 1 when income_quartile == 1 and 0 otherwise, and then include that variable in the regression (e.g., using mutate(income_quartile == 1)),
use fixest’s built-in i function inside of the regression where you define the reference level (the level to be dropped, here: 4) e.g., y ~ x + i(income_quartile, ref = 4).
The estimated coefficient on pct_smoke is -3.869: moving from no smoking to 100-percent smoking is associated with a 3.869-year decrease in life expectancy, holding all else fixed.
[05] Suppose we want to use the local tax rate tax_rate and local tax progressivity tax_prog as instruments.
Estimate the first-stage regression using tax_rate and tax_prog as instruments for pct_smoke—keeping the same controls as in [04]. Use heteroskedasticity-robust standard errors.
Do the two tax variables appear relevant? Report the results of the regression and explain your answer about the relevance of your instruments.
Bonus: Use the wald function from fixest to jointly test the significance of the two variables.
Wald test, H0: joint nullity of tax_rate and tax_prog
stat = 13.6, p-value = 1.277e-6, on 2 and 3,098 DoF, VCOV: Heteroskedasticity-robust.
Both tax variables have negative coefficients, which is consistent with the idea that higher taxes reduce smoking. Hypothesis tests for each of the variables reject zero at conventional levels; the joint test also reject zero.
Thus, we have statistically significant evidence that our tax instruments are relevant for smoking incidence at the county level.
[06] Estimate the reduced-form regression by replacing pct_smoke in [04] with the two tax variables.
Report your results.
Do we need exogeneity to interpret the reduced-form coefficients as causal effects of the tax variables on life expectancy?
How do you interpret the reduced-form coefficients?
The reduced-form coefficients on tax_rate and tax_prog tell us that higher tax rates and high tax progressivity are both associated with higher levels of life expectancy—holding the listed controls fixed.
Under the IV logic, this reduced-form effect works through smoking.
In practice, we need the exogeneity assumption to interpret the reduced-form coefficients as causal effects of the tax variables on life expectancy. If local tax policy affects life expectancy through channels other than smoking (or even correlates with other variables that affect life expectancy), then the reduced-form coefficients are not causal effects of the tax variables.
[07] Estimate the causal effect of smoking on life expectancy using two-stage least squares (2SLS), instrumenting pct_smoke with taxrate and tax_st_diff_top20. Keep the same controls as in [04] and use heteroskedasticity-robust standard errors.
Report your 2SLS results.
You may use feols from fixest, iv_robust from estimatr, or any other package you like.
[08] Compare your OLS and 2SLS (IV) estimates for smoking. Which one would you trust more for causal inference here, and why? Be explicit about 2SLS’s assumptions.
Answer OLS and IV give very different estimates. While both suggest a negative association between smoking and life expectancy, the 2SLS estimates suggest a much larger negative effect—the 2SLS estimate for smoking is -28.627, which is more than 7 times the size of the OLS estimate.
If I had to choose, I would only treat the 2SLS estimate as potentially closer to a causal effect, because 2SLS is designed to isolate exogenous variation in smoking. But I would still be cautious. Exogeneity is the key issue: if local tax policy affects life expectancy through channels other than smoking—for example, through public spending, income, migration, or other health-related behaviors—then the 2SLS estimate is not causal. So I would say 2SLS is more credible than OLS only if I can defend exogeneity. That defense is not obvious here.
[09] Beyond exogeneity issues, the relevance of your instrument(s) is a huge concern when using instrumental variables or two-stage least squares. Use the probability limit of the instrumental variables estimator to discuss how a weak instrument could substantially bias the estimator. \[\text{plim} \hat{\beta}^\text{IV} = \beta + \dfrac{\text{Cov}(z,\, u)}{\text{Cov}(z,\, x)}\] for instrument \(z\), endogenous regressor \(x\), and disturbance \(u\).
[10]On the topic of weak instruments: We should probably account for the fact that our disturbance likely correlates within state.
Re-run the first-stage regression but with standard errors that are robust to the disturbance to correlate (cluster) at the state level (e.g., cluster = ~state.abb).
Do the tax variables still appear relevant? Report your results and explain your answer.
4 Section 2: Final Review
This section is mostly conceptual. Keep your answers concise but precise.
[11] Define the concept of exogeneity in the context of regression analysis. Why is it important for causal inference?
Answer Exogeneity means that the regressors are uncorrelated with the error term in the regression. More formally, exogeneity says \(E[u | x] = 0\). Exogeneity is crucial for causal inference because it ensures that the estimated coefficients reflect the causal effect of the regressors on the outcome, rather than being biased by omitted variables, reverse causality, or other confounding factors.
[12] In one or two sentences each, define
a population model,
a sample model,
a parameter,
an estimator,
an estimate,
a standard error.
Answer
A population model describes the true data-generating process in the population.
A sample model is the model we estimate using observed sample data.
A parameter is a numerical characteristic of the population model that we want to learn about.
An estimator is a rule or formula that turns sample data into an estimate.
An estimate is the realized numerical value produced by an estimator in one sample.
A standard error measures the sampling uncertainty of an estimate.
[13] Explain the two conditions required for omitted-variable bias. If the omitted variable increases \(y\) and is positively correlated with an included regressor \(x_j\), what direction is the bias in the OLS estimate of \(\beta_j\)?
Answer Omitted-variable bias requires two things: (1) the omitted variable must affect the outcome, and (2) it must be correlated with an included explanatory variable. If the omitted variable increases \(y\) and is positively correlated with \(x_j\), then the bias in the OLS estimate of \(\beta_j\) is upward.
Interpret \(\beta_1\), \(\beta_2\), and \(\beta_3\).
Answer
\(\beta_1\) is the difference in expected wage between women and men when \(school = 0\).
\(\beta_2\) is the return to one more year of schooling for men, the omitted group.
\(\beta_3\) is the difference in the return to schooling between women and men.
[15] Suppose heteroskedasticity is present, but the exogeneity assumption still holds. What happens to
the OLS coefficient estimates,
the usual OLS standard errors,
OLS efficiency?
Answer
The OLS coefficient estimates remain unbiased and consistent.
The usual OLS standard errors become biased, so the usual t tests and confidence intervals are unreliable.
OLS is no longer the most efficient linear unbiased estimator.
[16] Give the best econometric tool for each setting below and explain why in one sentence.
You believe OLS coefficients are fine, but you do not know the form of the heteroskedasticity.
Observations may be correlated within states, classrooms, or firms.
You want to correct your standard errors because the distrubance correlates with itself through time.
You want a heteroskedasticity test that is less tied to a specific functional-form assumption.
Answer
Use heteroskedasticity-robust standard errors because they fix inference without requiring you to know the exact variance function.
Use clustered standard errors because the problem is within-group correlation, not just heteroskedasticity.
Use Newey-West standard errors because they allow for autocorrelation in the disturbance.
Use the White test because it is more flexible about the form of heteroskedasticity than a more structured test such as Goldfeld-Quandt.
[17] What does it mean for an estimator to be consistent? Can an estimator be biased but consistent? Can an estimator be unbiased but inconsistent?
Answer An estimator is consistent if it converges in probability to the true parameter as the sample size grows. Yes, an estimator can be biased but consistent if the bias vanishes as the sample gets large. Yes, an estimator can be unbiased but inconsistent if it is centered correctly in every sample but does not converge to the true value as the sample size grows.
[18] What happens to the OLS slope estimate when an explanatory variable is measured with classical measurement error? Why is this often called attenuation bias?
Answer Classical measurement error in an explanatory variable biases the OLS slope toward zero. It is called attenuation bias because the noise weakens the observed relationship between x and y, shrinking the estimated effect in magnitude.
[19] How does autocorrelation affect OLS in
a static time-series model, and
a dynamic model with a lagged dependent variable?
Answer
In a static time-series model, autocorrelation typically leaves the coefficients unbiased and consistent if the relevant exogeneity assumption holds, but it biases the usual standard errors.
In a dynamic model with a lagged dependent variable, autocorrelation can violate exogeneity and make OLS biased and inconsistent.
[20] Define stationarity. Why can non-stationary series generate spurious regressions? Give
one case where differencing is appropriate,
the null hypothesis in a Dickey-Fuller or augmented Dickey-Fuller test.
Answer
A stationary process has a stable mean, variance, and autocovariance structure over time.
Non-stationary series can drift together over time even when they are unrelated, which can create misleadingly significant regression results.
Differencing is appropriate when the series is difference-stationary, such as a random walk.
In a Dickey-Fuller or augmented Dickey-Fuller test, the null is that the series has a unit root, meaning it is non-stationary.
