Homework 5: Work, Study, Enroll (AK)

Purpose

The objective of this homework is to practice reading, interpreting, and critiquing a published paper that uses instrumental variables (IV) — This homework trains you to be the kind of analyst who can pick up a paper, understand its design, identify its limits, and credibly communicate its findings to a decision-maker who has never heard of a LATE. This homework was inspired by work on APP, so think of this as something you may run into while doing your APP.

Guidelines

• You can work by yourself or with groups of up to two.
• Submit your group answers to Gradescope (within Canvas). One submission per group, please.
• You will get points for correct answers. Points will be deducted if the answer contains more information than necessary, or if it contains incorrect statements alongside correct ones. We reward precision: use the fewest characters needed to maximize accuracy. This will get more strict over time.
• Your responses should be professionally formatted and written.
• The due date is March 27th, at 9pm EDT.
• You can answer quantitative questions to the nearest 0.01.

Preamble

You work as a research analyst at a policy organization advising a state legislator in Central Appalachia — a region where community college (CC) enrollment rates are well below national averages and limited access to credentials constrains economic mobility. Your boss has been asked a direct question:

"We have $2 million to expand access to community college. Should we spend it on Federal Work-Study offers? And if so, how many students would we reach?"

Ideally, you can look through this paper and see if you can find the answer straight up. This homework will walk you through steps to get there, but I think it’s good practice to see if you can do it without the guardrails, I won’t always be next to you! Anyways, To answer this, your organization has identified a recent paper: Minaya, Scott-Clayton & Soliz (2026), "The Causal Effects of Federal Work-Study Offers on College Enrollment and Program Participation" (EdWorkingPaper No. 26-1400). The paper studies 66,360 FAFSA applicants at a large multi-campus public college system, exploiting quasi-experimental variation in who receives FWS offers based on the timing of FAFSA submission relative to campus-specific budget cutoffs. Notice that this paper’s main strategy is difference-in-difference, which we haven’t studied yet. However, they also have a version of their strategy that is IV, the questions about IV refer to that part of the strategy.

Your task: evaluate the evidence, interpret the design, and write rigorous answers that could form the basis of a policy memo to the legislator.

Section 1: Understanding the Research Design

1. The natural experiment. In two to three sentences, describe the source of variation the authors exploit. What makes some students eligible for FWS offers, and what makes when they file their FAFSA matter? Do not use the word "instrument" yet — just explain the setting plainly.
‣

Answer

✅

Each campus receives a fixed, limited FWS budget at the start of the aid year. Students who file their FAFSA before the date when the campus budget runs out are eligible to receive FWS offers; those who file after are not. Because students do not know the exact cutoff date in advance, filing timing relative to that cutoff is plausibly independent of a student's underlying enrollment intentions — creating a natural experiment in who receives an offer.

2. The IV framework. Let $Z_i$ be the instrument, $D_i$ the treatment, and $Y_i$ the outcome (pick one outcome of interest to your boss).
1. Define each variable precisely as used in this paper and also explain them in simple terms so that your boss can understand.
‣
Answer
✅
• $Z\_i = \text{Eligible}\_i \times \text{Before}\_i$ : equals 1 if the student is FWS-eligible AND filed FAFSA before the campus budget cutoff.
• $D\_i$ : received a FWS offer (= 1) or not (= 0).
• $Y\_i$ : Fall 2017 enrollment indicator (= 1 if enrolled).
2. Write the first-stage, reduced-form, and structural (second-stage) equations. Some may be written in the paper and some may not. Use these equations to also express the IV estimator as a ratio. Notice the two important “covariates” this IV needs.
‣

Answer

✅

First stage:

$D_i =\pi_0 + \pi_1 Z_i + X_i'\delta + \nu_i$

Reduced form:

$Y_i = \gamma_0 + \gamma_1 Z_i + X_i'\lambda + \varepsilon_i$

Structural equation:

$Y_i = \alpha_0 + \alpha_1 D_i + X_i'\beta + u_i$

IV estimator: $\hat{\lambda_1^{IV}} = \frac{\hat{\gamma_1}}{\hat{\pi_1}}$ — reduced form divided by first stage.

3. Understand the results part 1
1. What does Figure 2 visually demonstrate?
2. Which table provides evidence of the “first stage”?
3. Interpret the of 0.178 from Table 3. (Notice that the layout of the table is a little bit different from what we are used to)
4. Is the instrument strong? Cite the relevant statistic from the paper.
‣

Answer

✅

(a) Figure 2 shows a sharp, discontinuous jump in FWS offer rates right at the campus budget cutoff date. Students filing just before the cutoff have substantially higher FWS offer rates than students filing just after, with smooth trends on both sides. This graphically confirms the first stage.

(b) Table 3

(c) Filing before the campus cutoff while eligible increases the probability of receiving a FWS offer by 17.8 percentage points. This quantifies how strongly the instrument shifts whether a student receives treatment.

(c) Yes. The F-statistic on the instrument is approximately 607–620 in Table 3, far exceeding the conventional threshold of 10 and stricter modern benchmarks. This is an exceptionally strong instrument; weak instrument bias is negligible.

4. Table 2 question
1. What is the purpose of a Table 2 in this IV design?
2. Table 2 shows a 2 percentage point imbalance in dependency status at the cutoff. Is this a serious concern? Explain.
‣

Answer

✅

(a) Balance checks test whether students filing just before the cutoff are statistically comparable on predetermined characteristics to students filing just after. If the instrument is valid — filing timing is as good as random near the cutoff — there should be no systematic differences in background characteristics across the threshold.

(b) The 2pp imbalance in dependency status is small in magnitude, borderline in significance, and the authors include it as a control in all regressions. Finding one marginal imbalance out of many tested variables is expected by chance. This is not a serious concern, especially given the corroborating evidence from Appendix Figure A2, which shows no discontinuities in other aid types at the cutoff.

Section 2: Interpreting the Results

5. What is the reduced-form effect on Fall 2017 enrollment for the full sample? (find the right table) Is it statistically significant? Economically significant?
‣

Answer

✅

The reduced-form effect is +1.0 percentage point (p = 0.15), which is not statistically significant. Averaging across all FAFSA applicants, the FWS offer does not detectably affect enrollment. This likely reflects genuine heterogeneity: enrollment effects are concentrated among specific subgroups, and averaging across all applicant types dilutes the signal to insignificance. It is not economically significant because enrollment before is about 73% so this is a small change.

6. For the CC applicant subsample, (find the right table) report: (a) the reduced-form effect on enrollment and its statistical significance, and (b) the first-stage coefficient.
‣
Answer
✅
For CC applicants:
• (a) Reduced-form effect on Fall 2017 enrollment: +4.1 percentage points (p < 0.01).
• (b) First-stage coefficient: +11.2 percentage points in FWS offer probability.
7. Using only the two numbers from the previous questions, compute the IV (LATE) estimate. Show your work clearly.
‣

Answer

✅

$\widehat{text{LATE}} = \frac{\text{Reduced form}}{\text{First stage}} = \frac{4.1 \text{ pp}}{11.2 \text{ pp}} = \frac{0.041}{0.112} \approx 0.366$

The IV estimate is approximately 36 percentage points (or 0.366 in decimal).

8. Interpret your estimate in: (a) precise technical language and (b) plain language for the legislator. No more than three sentences each.
‣

Answer

✅

(a) Technical: For community college applicants who are compliers — those who received a FWS offer because they filed before the campus budget cutoff but would not have received one had they filed after — receiving a FWS offer increases the probability of Fall 2017 enrollment by approximately 36 percentage points.

(b) Plain language: Among CC students whose chance of getting a work-study offer depended on whether they happened to file their FAFSA before or after the campus ran out of money, getting an offer dramatically increased the chance they enrolled in the fall. About 36 out of every 100 students in this group who received an offer enrolled who otherwise would not have.

9. Using the paper's specific design: (a) describe who the compliers are in concrete terms, and (b) describe two groups of non-compliers — always-takers and never-takers — in this specific context.
‣

Answer

✅

(a) Compliers are CC applicants who filed their FAFSA close to the campus budget cutoff, such that they received a FWS offer only because they filed before funds ran out — had they filed slightly later, they would not have received one. Their offer receipt is driven by the luck of timing relative to the cutoff.

(b) Non-compliers:

• Always-takers: Students who receive a FWS offer regardless of when they file — e.g., those who apply very early in the year, well before any budget risk.
• Never-takers: Students who would not receive a FWS offer regardless of timing — e.g., those who do not meet eligibility criteria no matter when they file.

The LATE is identified only for compliers and says nothing about the effect of FWS offers on always-takers or never-takers.

Section 3: The Policy Application — "The APP Problem" (Translating estimates into actionable information)

10. As your client's aide you have these two number to report: the reduced-form effect and the IV estimate. She asks which one to use to evaluate her proposed FWS expansion. Which do you recommend, and why? Be precise about what each quantity measures.
‣

Answer

✅

Use the LATE (~36pp).

The LATE (~36pp) is the causal effect of receiving a FWS offer on enrollment, for compliers. Since the legislator's policy is to direct offers to students and she wants to know what receiving an offer does to enrollment, the LATE answers her question directly.

11. The legislator wants 400 additional community college enrollees via FWS offers. Using the IV estimate, how many FWS offers must she make? Show your work.
‣

Answer

✅

$\text{Offers needed} = \frac{\text{Target new enrollees}}{\text{IV estimate}} = \frac{400}{0.36} \approx 1{,}111 \text{ offers}$

The legislator would need to make approximately 1,111 FWS offers to generate 400 additional community college enrollees, assuming the complier composition and effect size in her region is comparable to the study population.

12. The study you are reading has a different population than Appalachian, so in order to understand if these are “larger” or “smaller” than what would be for this population, Identify two specific ways Central Appalachian CC students may differ from the study population. For each, state whether the true LATE in that context would be larger or smaller than 36 percentage points and explain your reasoning. (Hint: which concept we’ve seen in class can help you talk about this?)
‣

Answer

✅

Many valid answers. Two examples:

1. Greater financial need — LATE likely larger. Central Appalachian students may face more acute liquidity constraints than students at the large urban system studied. For these students, a FWS offer is a proportionally more valuable income opportunity on the margin of enrollment. The enrollment response to an offer would be larger, so true LATE > 36pp.

2. Weaker job-placement infrastructure — LATE likely smaller. The study institution likely has established systems to match FWS recipients with jobs. Smaller Appalachian campuses with fewer available positions may see lower conversion from offer to actual placement, weakening the causal chain from offer to enrollment. True LATE < 36pp.

13. Now let's go back to the original question, we think you are ready for it: "We have $2 million to expand access to community college. If we spend it on Federal Work-Study offers? And if so many students would end up enrolled?"
‣
Answer
✅
The paper reports that FWS students earned an average of $2,000 over the course of the year. Using this as the cost per offer, the legislator could extend 1,000 FWS offers:
$\text{Number of offers} = \frac{\$2{,}000{,}000}{\$2{,}000} = 1{,}000$
Applying the IV (LATE) estimate of ~0.366:
$\text{New enrollees} = 1{,}000 \times 0.366 \approx 366$
This means the legislator could reach roughly 366 additional community college enrollees — students who would not have enrolled without the offer.
Key caveats:
• The LATE is identified for compliers; this calculation assumes new offer recipients resemble the study's complier population.
• The $2,000 average earnings figure comes from the study institution; actual FWS award sizes in Central Appalachia may differ.
• The 36pp estimate already incorporates the fact that many offer recipients never take a FWS job (see Q17) — no further adjustment for take-up is needed.
• As discussed in Q12, local differences in financial need and institutional capacity could push the true effect higher or lower than 36pp.
14. A cost effectiveness analysis provides a ratio, in this setting you want to know the cost per student enrolled in community college of this policy, given the numbers you just obtained what would be that estimate for this policy? Notice that once we have this number we can compare them to other policy options to see what's the best "bang for your buck". This is a prime example of how research informs policy making.
‣

Answer

✅

From Q13, we estimated that $2 million in FWS offers yields approximately 366 additional CC enrollees. The cost-effectiveness ratio is:

$\text{Cost per enrollee} = \frac{\text{Total cost}}{\text{Additional enrollees}} = \frac{\$2{,}000{,}000}{366} \approx \$5{,}464 \text{ per student}$

That is, each additional community college enrollment induced by an FWS offer costs roughly $5,464.

This figure can now be compared to other enrollment-boosting policies — e.g., Pell Grant expansions, tuition waivers, or outreach campaigns — to determine which delivers the most enrollees per dollar. A policy with a lower cost per enrollee is more cost-effective. This is the core logic of cost-effectiveness analysis: it translates research estimates into a common metric that lets decision-makers compare apples to apples across policy options.

Section 4: IV Assumptions

15. State the exclusion restriction for this paper's instrument in plain English. (b) What did the authors test in Appendix Figure A2, and what did they find? Why does this test matter?
‣

Answer

✅

(a) The exclusion restriction requires that the instrument — being FWS-eligible and filing before the campus budget cutoff — affects fall enrollment only through its effect on receiving a FWS offer. The instrument should not affect enrollment through any other channel, such as by changing Pell Grant amounts, loan eligibility, or state aid.

(b) Appendix Figure A2 tests whether Pell Grant receipt, state grant aid, and loan receipt show any discontinuity at the campus budget cutoff. The authors find no significant effects on any of these other aid types. This rules out the most plausible violation of the exclusion restriction: if the instrument were also affecting other financial aid, the IV estimate could not be cleanly interpreted as the effect of FWS offers alone.

16. The paper shows that receiving a FWS offer increases the probability of holding a FWS job by about 27 percentage points — far less than 100%. Why don't the authors use $Z_i$ to estimate the causal effect of actually working a FWS job on enrollment? What assumption would this violate?
‣

Answer

✅

Using $Z_i$ to instrument for actual FWS job-holding rather than offer receipt would violate the exclusion restriction.

If we redefine the treatment as "holds a FWS job," we require that $Z\_i$ affects enrollment only through job-holding. But this is implausible: receiving a FWS offer may itself affect enrollment through channels that do not involve taking the job — by reducing financial uncertainty, signaling affordability, or motivating enrollment finalization. These are direct paths from $Z\_i$ to $Y\_i$ that bypass the redefined treatment. Once those paths exist, the exclusion restriction is violated and the IV estimate is not identified as the effect of job-holding.

In short: offers affect enrollment both through taking the job and through non-job channels. This instrument cannot isolate only the job-taking path.

17. The legislator says: "If only about 27% of offer recipients actually get a FWS job, doesn't that mean we need to scale up our 1,111 estimate even further — to account for all the offers that won't be converted into jobs?" How do you respond?
‣

Answer

✅

No — the 1,111 figure does not need to be scaled further for take-up. The IV estimate of ~36pp already incorporates the full behavioral chain from receiving an offer to enrolling — including the fact that many offer recipients do not take a FWS job.

The reduced form measures the total effect of the offer on enrollment through all channels (job-taking, reduced financial anxiety, etc.). The first stage measures how the instrument shifts offer receipt. Their ratio — the LATE — is the causal effect of the offer on enrollment, not the effect of the job. The 27% take-up rate describes a mechanism; it is not a correction factor.

Dividing by take-up again would implicitly assume that enrollment gains flow only through job-holding — but the IV estimate already reflects reality, including students who enroll after receiving an offer without ever taking the job. The calculation stands.

Section 5: Doing it with data

18. The results before were obtained from reading the paper. When you don’t have access to data, it is important to rely on paper’s estimates to try to answer the questions as best as one can. There are other opportunities in which you will have the data readily available and could run some regressions. The data used in this paper is of restricted access, but we are providing you with simulated data created with the idea to replicate the results (they won’t be exactly the same as the paper, but more than being exactly the same the question is whether you can replicate the empirical exercise with data). Use the data and the information provided in the paper of the models to create a do-file that finds the following regressions of interest. For the purposes of these questions our main outcome of interest is enrollment in Fall 2017 for the full sample and sub-sample of community college applicants. Present a single table that provides the following regressions, generate a variable called z or instrument for the instrument:

Data can be found here (fws_simulated.dta)

• The regression of the first-stage from the model with covariates, for the full sample.
• The regression of the ITT from the model with covariates, for the full sample.
• The regression of the first-stage from the model with covariates, for the sample of community college applicants.
• The regression of the ITT from the model with covariates, for the sample of community college applicants.
• The regression that represents the causal effect of receiving a FWS offer on Enrollment in Fall, 2017 for the sample of community college applicants. (Hint: this should be very very similar as the wald estimator version if you wanted to check)

‣

Answer

Extensions (non graded - but good practice)

19. Table 4 shows large enrollment effects for CC applicants (+4.1pp) and for independent students, but null effects for four-year applicants and dependent students. Provide two distinct economic explanations for this pattern.
‣

Answer

✅

Many valid answers. Two examples:

1. Marginal students are concentrated among CC applicants. The LATE is estimated for compliers — students whose enrollment decision is sensitive to whether they get a FWS offer. CC applicants tend to face more uncertain enrollment decisions (financially precarious, weighing work vs. school trade-offs) than four-year applicants who have made stronger pre-existing enrollment commitments. More marginal students among CC applicants means a larger LATE for that group.

2. Financial need and liquidity constraints. Independent students must self-finance their education; a FWS offer represents a meaningful income supplement that directly eases the cost of attending. Dependent students with parental support face lower financial pressure, so the marginal value of the offer is smaller and their enrollment decision is less sensitive to it.

20. The paper concludes that "expanding funding alone is unlikely to deliver full benefits without infrastructure to match students to jobs." Using take-up rates and IV logic, explain what the authors mean and propose one concrete policy complement to a FWS funding expansion.
‣

Answer

✅

The IV estimate captures the enrollment effect of a FWS offer, which is substantial. But the low take-up rate — only ~27pp of offer recipients actually hold a FWS job — means that many students enroll without accessing the work component. The long-run benefits of FWS (income during school, work experience, professional development) depend on actually holding the job, not merely receiving the offer. Expanding funding without improving job-matching generates enrollment gains that do not fully translate into the program's intended downstream benefits.

One concrete complement: Fund dedicated FWS job-placement coordinators at community colleges — staff who proactively match eligible students with available positions, help navigate scheduling conflicts, and follow up with offer recipients who have not yet been placed. This increases take-up, ensuring enrollment gains translate into fuller program participation and its associated benefits.