Problem Set 4.3 — Regression Discontinuity: Local-Polynomial Fitting and Bandwidth Experiments
Covers Ch 4.3. Methods allowed: everything through Ch 4.3 — the sharp-RD estimand \tau_{\text{SRD}}=\lim_{x\downarrow c}\mathbb{E}[Y\mid X=x]-\lim_{x\uparrow c}\mathbb{E}[Y\mid X=x] and the continuity / local-randomization assumption; local linear estimation with a window of half-width h (the bandwidth) and a triangular kernel; the bias–variance trade-off in h and why local-linear beats a global high-order polynomial (Gelman–Imbens 2019); the data-driven bandwidth choices of Imbens–Kalyanaraman (2012) and the robust bias-corrected inference of Calonico–Cattaneo–Titiunik (2014); fuzzy RD as the Wald ratio \tau_{\text{FRD}}=(\text{jump in }Y)/(\text{jump in }D) with the above-cutoff indicator Z_i=\mathbf 1\{X_i\ge c\} as the instrument; the McCrary (2008) density test for manipulation; and the validity-threat checklist of §9 (manipulation, covariate imbalance, compound treatments, donut, bandwidth sensitivity). You may treat OLS, the Wald estimator and LATE/compliers (Week 3), and weak-instrument logic (Ch 3.5) as known. Show your reasoning; a boxed number with no argument earns no credit, and whenever you invoke an assumption you must name it — “continuity,” “relevance,” “exclusion,” “monotonicity” — never just “the assumption.”
Total: 100 points. Six problems, escalating. Problem 1 is conceptual (state the sharp-RD estimand and the continuity assumption, and contrast it with CIA). Problem 2 is a by-hand local-linear RD: fit a line on each side of the cutoff and read the jump. Problem 3 is the bandwidth bias–variance problem (what happens as h shrinks and grows; why local linear, not a global quartic — Gelman–Imbens). Problem 4 is a fuzzy-RD problem (jump in treatment probability \to Wald ratio, connected back to Week-3 IV). Problem 5 is a McCrary-density/manipulation problem (what sorting does and how the density test detects it). Problem 6 is a validity-threats critique (compound treatments at the same cutoff).
One idea runs through every problem, so hold it from the start. RD reads a causal effect off a jump: the vertical gap in \mathbb{E}[Y\mid X] exactly at the cutoff c. Continuity says that, absent the treatment, the outcome would have passed through c smoothly, so the gap can only be the treatment. Local linear regression is just the honest way to read the height of each side’s curve at c; the bandwidth is how wide a window you trust; fuzzy RD divides the outcome jump by the treatment jump exactly as the Week-3 Wald estimator did; and the validity tests all ask the same single question — is the jump really the treatment, or is something else also jumping at c?
We use the chapter’s two running cutoffs: Sam’s Russell 1000/2000 reconstitution-rank cutoff (recentered so c=0, with X\ge 0 landing in the Russell 2000) and Maya’s lender credit-score cutoff at 660.
Problem 1 — State the estimand and the identifying assumption (14 points)
Maya studies an automated lender that approves every applicant with a credit score X_i\ge 660 and rejects everyone below, with the loan officer having no discretion. The treatment is loan approval, D_i=\mathbf 1\{X_i\ge 660\}, and the outcome Y_i is the applicant’s two-year financial-distress index (lower is better). She wants the causal effect of approval.
(a) (3 pts) Write the sharp-RD estimand \tau_{\text{SRD}} for Maya’s design as a difference of two one-sided limits, in the notation of Ch 4.3. In one sentence, say in plain words what each limit is and why their difference is a treatment effect rather than just “the slope of Y in X.”
(b) (4 pts) State the continuity assumption formally, in potential-outcomes language (Y_i(1),Y_i(0)), and translate it into one plain-English sentence about what would have happened to a barely-approved applicant had the cutoff sat slightly higher. Which of the two potential-outcome functions is the one we never directly observe on the treated side, and which side of the cutoff supplies it?
(c) (4 pts) The chapter calls RD’s assumption “weaker than CIA, and partly checkable.” Explain, in two or three sentences, the precise difference: what does CIA (Week 3) demand, what does continuity demand instead, and why does that make continuity easier to defend?
(d) (3 pts) Maya’s estimate \tau_{\text{SRD}} is the effect of approval for whom? Name the population, and say in one sentence why this “local” estimand is nonetheless exactly the right number for a regulator deciding whether to move the cutoff from 660 to 650.
Problem 2 — A by-hand local-linear RD (20 points)
Sam bins stocks by reconstitution-rank distance from the cutoff (c=0; X\ge 0 is the Russell 2000, treated) and computes mean next-year passive-fund ownership (in percentage points) in each bin. Working within a bandwidth of h=4, he keeps the four bins inside the window and reads off the bin midpoints:
| Bin midpoint x | Side | Mean passive ownership \overline Y (%) |
|---|---|---|
| -3 | control (R1000) | 5.2 |
| -1 | control (R1000) | 5.4 |
| +1 | treated (R2000) | 5.9 |
| +3 | treated (R2000) | 6.1 |
Treat each side as exactly two points, so the local-linear fit on each side is just the straight line through its two points.
(a) (6 pts) Fit the control-side line Y=a_- + b_-\,x through (-3,5.2) and (-1,5.4). Compute the slope b_- and intercept a_-, then the extrapolated control-side height at the cutoff, \lim_{x\uparrow 0}\widehat{\mathbb{E}}[Y\mid X=x]=a_-. Show the two-point slope calculation.
(b) (6 pts) Fit the treated-side line Y=a_+ + b_+\,x through (+1,5.9) and (+3,6.1). Compute b_+, a_+, and the extrapolated treated-side height at the cutoff, \lim_{x\downarrow 0}\widehat{\mathbb{E}}[Y\mid X=x]=a_+.
(c) (4 pts) Report \hat\tau_{\text{SRD}}=a_+-a_-. Explain in one sentence why this is not simply the difference of the two nearest bins (5.9-5.4=0.5), and what the local-linear fit corrects for.
(d) (4 pts) Sam considers replacing the local-linear fit with the crude “nearest-bin difference” 5.9-5.4. State which estimate is bigger, and explain — using the direction of the side-trends you found in (a) and (b) — whether the nearest-bin shortcut is biased up or down relative to the local-linear jump, and why. (No triangular-kernel weighting is needed here because each side has only two equidistant points; state in one line why the kernel would not change a two-point line.)
Problem 3 — Bandwidth, bias, variance, and the global-polynomial trap (18 points)
This problem uses the verified simulation of Ch 4.3 §11. The data-generating process has a true jump of \tau=0.6; the smooth part of \mathbb{E}[Y\mid X] is flat-ish below the cutoff but convex (curving upward) on the treated side. The chapter reports these local-linear estimates:
| Bandwidth h | \hat\tau | \widehat{\text{se}}(\hat\tau) |
|---|---|---|
| 4 | 0.56 | 0.18 |
| 32 | 0.41 | 0.07 |
and, for contrast, a global quartic fit to all 6{,}000 points reports \hat\tau\approx 0.41 with a tight-looking standard error.
(a) (5 pts) Explain the bias–variance trade-off across the two bandwidths using the table’s own numbers. As h goes from 4 to 32: which way does the standard error move, and why (in terms of how many points each fitted line uses)? Which way does the bias move, and why (in terms of the curvature on the treated side)?
(b) (3 pts) At h=4 the estimate is close to the truth (0.56 vs 0.60) but the confidence interval \hat\tau\pm 1.96\,\widehat{\text{se}} is wide; at h=32 the interval is narrow but mis-centered. Compute both naive 95\% intervals and state, in one sentence, which window an honest analyst should prefer and why “narrow” is not the same as “accurate.”
(c) (5 pts) The global quartic returns essentially the same biased point estimate (0.41) as the too-wide local-linear fit, but with a deceptively tight standard error. Give the Gelman–Imbens (2019) argument in three sentences: name the three reasons a high-order global polynomial mis-estimates a boundary value, and say what the “deceptively tight standard error around a wrong number” reveals about the difference between precision and accuracy.
(d) (5 pts) Name the two data-driven bandwidth procedures from the chapter and state precisely what each delivers: (i) what quantity does Imbens–Kalyanaraman (2012) minimize, and what does that imply about whether their optimal h leaves any bias behind? (ii) Given that the IK/MSE-optimal bandwidth deliberately leaves a non-negligible bias, what two things does Calonico–Cattaneo–Titiunik (2014) robust bias-correction do to repair the confidence interval, and what coverage problem does it fix?
Problem 4 — Fuzzy RD as the Wald ratio (18 points)
In modern Russell data the design is fuzzy: crossing the rank cutoff does not force a fixed amount of passive ownership; it raises the probability and intensity of heavy Russell-2000-tracking ownership. Sam defines the treatment D_i=1 as “heavy passive uptake” and estimates four boundary limits (each read off a local-linear fit, h fixed):
| Boundary limit | Value |
|---|---|
| \lim_{x\downarrow 0}\mathbb{E}[D\mid X=x] (treated-side treatment share) | 0.70 |
| \lim_{x\uparrow 0}\mathbb{E}[D\mid X=x] (control-side treatment share) | 0.30 |
| \lim_{x\downarrow 0}\mathbb{E}[Y\mid X=x] (treated-side outcome, governance index) | 61.0 |
| \lim_{x\uparrow 0}\mathbb{E}[Y\mid X=x] (control-side outcome, governance index) | 57.4 |
The outcome Y is a firm-governance index (higher = more shareholder-friendly).
(a) (4 pts) Compute the jump in treatment (the first stage) and the jump in outcome (the reduced form) at the cutoff. Name the condition that the treatment jump is the empirical content of, and say whether 0.40 is reassuring or worrying on that score.
(b) (4 pts) Compute the fuzzy-RD estimand \hat\tau_{\text{FRD}} as the ratio of the two jumps. Write the ratio explicitly in the form (jump in Y)/(jump in D), and interpret the number: what does heavy passive ownership do to the governance index, and for which firms?
(c) (4 pts) Identify the instrument Z_i in this design and map each piece of the ratio to its Week-3 name. Which jump is the “first stage,” which is the “reduced form,” and why is \hat\tau_{\text{FRD}} literally the Wald estimator from Ch 3.4?
(d) (3 pts) State what exclusion means here, in RD language: what must crossing the rank cutoff do (and not do) to the governance index? Give one concrete channel that, if real, would violate it — and note which §9 threat that channel is an instance of.
(e) (3 pts) Suppose FTSE Russell’s banding rules weakened, so the treatment jump fell from 0.40 to 0.05 while the outcome jump stayed near 3.6. Without recomputing precisely, say what happens to the fuzzy-RD point estimate and, more importantly, to its precision, and name the Ch 3.5 pathology this is an instance of.
Problem 5 — The McCrary density test and manipulation (16 points)
Maya worries that applicants (or loan officers helping them) can nudge a credit score from just below 660 to just above by paying down a small balance right before the pull — but only the more financially-capable applicants know to do this.
(a) (4 pts) Describe the McCrary (2008) density test in words: what object does it estimate on each side of the cutoff, what is the null hypothesis, and what does rejecting the null indicate? State what you hope to see (reject or fail to reject) and why.
(b) (4 pts) Explain the fingerprint of the specific manipulation Maya fears. If capable applicants sort from just-below to just-above 660, what happens to the histogram of credit scores in the bins immediately below and immediately above 660? Draw the link from “sorting” to “a jump in the density at c.”
(c) (4 pts) Explain why this sorting breaks the design even though approval still flips cleanly at 660. Use the local-randomization / continuity language: which group becomes “enriched,” on what kind of characteristic, and why are the just-above and just-below applicants no longer exchangeable? What does the resulting \hat\tau_{\text{SRD}} now mix together?
(d) (4 pts) Maya runs the test and fails to reject the null (the density looks smooth through 660). Her labmate says, “Great — the density test passed, so the design is clean.” Give the two-part correct response: (i) why a passed density test is necessary but not sufficient, and (ii) name one validity threat from §9 that a passed density test cannot rule out at all.
Problem 6 — Validity threats: the compound-treatment critique (14 points)
Maya finds a large, clean-looking jump in two-year default rates at the credit-score cutoff of 660: the density test passes, and a covariate-balance RD on pre-application debt-to-income shows no jump. She is ready to write “approval raises defaults by \hat\tau.” Then she learns that, at exactly 660, the lender’s underwriting system also (i) switches the loan from a fixed to a lower variable interest rate, and (ii) makes the loan eligible for securitization (which changes who services it).
(a) (4 pts) Name this validity threat from §9 and state, in one sentence, the single sentence that summarizes when RD is identified — the sentence this threat violates.
(b) (4 pts) Explain why no statistical test — not McCrary, not covariate balance, not a donut, not a bandwidth sweep — can detect this problem. Contrast it explicitly with the manipulation threat of Problem 5, which does leave a statistical fingerprint. What kind of evidence must Maya use instead?
(c) (3 pts) Reframe the situation honestly: Maya’s RD still estimates a real causal effect. Of what? Write the precise object her \hat\tau identifies, and explain why it is not the “effect of approval” she wanted.
(d) (3 pts) Connect this to fuzzy RD’s exclusion restriction (Problem 4). If Maya’s design were fuzzy, with the above-660 indicator as the instrument for approval, which IV condition does the compound treatment break, and through what mechanism does it contaminate the Wald ratio? (One or two sentences; name the condition.)
End of PS 4.3. Solutions in book/appendices/E-solutions-manual/E-w4-ps4.3-solutions.md. The hands-on companion is nb4.3 — “RD with rdrobust; CCT bandwidths”: there you re-do Problem 2’s hand-rolled local-linear fit against the package’s data-driven CCT bandwidth and robust bias-corrected interval, watch Problem 3’s bias–variance trade-off and the Gelman–Imbens global-polynomial pathology in live numbers, run the McCrary/Cattaneo–Jansson–Ma density test of Problem 5, and switch to the fuzzy design of Problem 4 to recover the effect via the IV/Wald ratio while precision collapses as the first-stage jump shrinks.