git:20260802.bfe550f to git:20260826.ef24590

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name: rdd

description: Design, estimate, validate, and write up a regression discontinuity analysis, in both the continuity and local-randomization frameworks, with the full falsification battery and a refusal rule for designs that fail validation. Produces advice with citations, R estimation and diagnostics code, and a drafted methods paragraph. TRIGGER on "regression discontinuity", "RDD", "RD design", "running variable", "cutoff", "threshold", "rdrobust", "bandwidth", "McCrary test", "density test", "fuzzy RD", "regression kink", or any setting where treatment switches at a known score threshold (loyalty tiers, spend thresholds, ranking cutoffs, algorithmic triggers, eligibility scores, age or tenure rules). Design triage across methods belongs to causal-design.

---

# Regression discontinuity

An opinionated RD workflow grounded in a read canon (references/canon.md, current as of

2026-07-28): the Cattaneo-Titiunik Annual Review of Economics survey and its applied companion,

the Cattaneo-Keele-Titiunik guide, which includes a real failed design this skill uses as its

refusal template. The deliverable is the specification decision with the citation that justifies

it, the estimation and diagnostics code in R (Stata and Python exist for the whole suite), and a

methods paragraph with the limitation stated in first person at the point of the choice.

Refresh path: run the litreview skill on the method since the canon date and fold results into

references/canon.md as flagged addenda.

## The design gate: is this an RD at all?

An RD requires a score, a known cutoff, and a treatment rule that existed ex ante and is

verifiable. Before any estimation, write the qualitative account: who computes the score, was

the cutoff public, can agents precisely control their score near it. Precise manipulation is the

most important threat. Lab-measured or third-party-computed scores resist it; self-influenced

scores (customer spend near a tier threshold, follower counts near a monetization bar) invite

exactly the sorting the density test detects, so the falsification battery carries more weight

in marketing settings than in the medical originals.

Two red flags, the first disqualifying on its own (from the failed Oncotype-DX application in

Cattaneo-Keele-Titiunik 2023):

1. Treatment take-up jumps at score values away from the official cutoff (the rule was soft:

reps contact leads below the threshold, managers grant status matches early). Plot take-up

against the score before anything else.

2. Covariate imbalance already in the smallest window around the cutoff: disqualifying when

the imbalanced covariate plausibly drives the outcome (the balance battery's standard,

below), otherwise a serious flag that demands an explanation before proceeding.

When a design fails, say so and decline to report an effect; the guide's own verdict on its

failed application ("the evidence does not support an RD analysis") is the template. Route the

question back to causal-design for another identification strategy.

## Triage: two frameworks, and which one leads

Sharp vs fuzzy is a fact of the institution, not a choice: sharp when the rule binds

mechanically (a trial expires, a discount ends at an anniversary), fuzzy whenever anyone can

cross their assignment (overrides, opt-ins). Under fuzziness always report both ITT effects (on

take-up and on the outcome) alongside the fuzzy ratio.

Continuity vs local randomization is decided by the score:

- Continuous score, many observations near the cutoff: continuity framework with local

polynomials leads; local randomization is the robustness complement.

- Discrete score (rule of thumb: roughly 30 or fewer distinct values) or very few observations

near the cutoff: local randomization leads. Continuity methods there extrapolate from the

nearest mass points and their effective sample size is the number of mass points, not the

number of observations. Discrete running variables (weeks of tenure, order counts, months

since signup) are the norm in marketing data, which makes this branch more common than the

econ literature suggests.

- Local randomization needs a strictly stronger assumption (potential outcomes unrelated to the

score inside the window), which must be argued, not assumed.

When both frameworks apply, run both; agreement is a robustness result, and the local

randomization CIs covering the continuity point estimate counts as consistency. Expect the

window to be far narrower than the bandwidth (in the guide's HIV application, 121 patients vs

2,593), so a local-randomization null alongside a significant continuity estimate can be a power

difference, not a contradiction.

## The consensus recipe (continuity framework)

Local linear regression, triangular kernel, MSE-optimal bandwidth, robust bias-corrected

confidence intervals (Calonico-Cattaneo-Titiunik). Report both the conventional and the robust

interval. The one-line justification: conventional 95 percent intervals at the MSE-optimal

bandwidth cover only about 80 percent, and bias correction with the matching variance adjustment

Hard rules from the review, stated as prohibitions because that is how it states them:

- Bandwidths must be data-driven and criterion-optimal; choosing one by hand "is discouraged."

MSE-optimal for the point estimate, CE-optimal when the interval is the object. Distinct

- Global polynomial fits are visualization only, never estimation (Gelman-Imbens): boundary

behavior, counterintuitive weighting, overfitting.

- Covariates are for precision only; they cannot restore identification of the canonical RD

parameter, and adjusting an invalid design changes the parameter rather than rescuing it. The

point estimate should barely move when covariates enter; a large move signals imbalance.

## The local-randomization recipe

Select the window by nested covariate-balance tests: start from the smallest window with at

least about 10 observations per side, enlarge until balance rejects, using a deliberately loose

threshold (p < 0.15, no multiplicity correction, since over-rejection just shrinks the window).

Then difference in means inside the window, with Fisherian randomization inference when the

window holds few observations (exact under the sharp null; point estimates and CIs then require

a constant-effect model), Neyman or super-population inference when it is well populated.

## Fuzzy designs: the IV discipline applies

The fuzzy estimand is a complier average effect at the cutoff under relevance, exclusion, and

monotonicity, so the iv skill's habits transfer:

- Test the first stage inside the bandwidth or window, never on the full sample; the full-sample

F overstates strength. The guide's contrast is the anchor: F around 698 in the valid design

against F = 1.51 in the failed one, where the first-stage effect is 0.15 with a Fisherian

p-value of 0.32. An in-bandwidth first-stage F that is neither the strong nor the hopeless

extreme goes to the iv skill's ladder: read it against the F targets there and report

Anderson-Rubin/CLR intervals rather than the 2SLS t.

- Argue exclusion qualitatively and concretely: it fails if crossing the cutoff changes behavior

through anything other than treatment (a low churn score triggering a retention call AND a

flag another team acts on).

- Monotonicity: safe when crossing the threshold moves treatment in one direction only; suspect

when overrides run in both directions (the failed-design pattern above, reps contacting

below-threshold leads while managers grant early crossings, is exactly the defiers case);

one-sided noncompliance buys it for free.

- Run fuzzy-ratio balance tests: instrument strength amplifies covariate bias, so imbalance

invisible to ITT balance can surface in the ratio.

- Use one MSE-optimal bandwidth for the ratio, not separate ones for numerator and denominator.

## Falsification battery

1. Qualitative manipulation account (who computes the score, who knows the cutoff).

2. Density continuity test (rddensity, robust bias-corrected) plus the exact binomial count

test in small windows (framework-free, works for discrete scores; keep the window narrow

enough that a constant assignment probability is sensible). A discontinuous density is

neither necessary nor sufficient for invalidity, but it demands an explanation; sorting can

be administrative rather than strategic.

covariate as the outcome, a fresh MSE-optimal bandwidth per covariate, robust p-values. A

failure on a covariate that plausibly drives the outcome invalidates the design. The

equivalence-test formulation (null of imbalance) is the more honest variant when you want to

claim balance affirmatively.

treatment effects do not contaminate the placebo.

original bandwidth, re-estimate. Binds hardest where agents can time their crossing (spend

warning sign; failure far above it is expected by construction and not damning.

Ex-post power calculations from observed effects are unreliable; when a null matters, report

minimum detectable effects instead (rdpower).

## The live dispute, carried honestly

Robust bias correction (this canon's school) vs honest uniform-in-bias inference

(Armstrong-Kolesar, Imbens-Wager; R package RDHonest). The honest school bounds the second

derivative by a constant M and gets uniformly valid intervals; the canon's objection is that a

data-driven M destroys the uniformity that motivates the method, and a manual M is equivalent to

choosing the bandwidth by hand. Default here: RBC. When a referee or coauthor asks for honest

intervals, report RDHonest alongside with the M choice justified in text, and cite both sides.

## Extensions, briefly

- Kink designs: same machinery on first derivatives; identification is more delicate.

- Multiple cutoffs or scores (tiered loyalty programs; geographic borders such as DMA

boundaries): cutoff-specific effects or normalize-and-pool (rdmulti), with the pooled

estimand's interpretation checked.

- RD in time: hard to justify as standard RD; the local-randomization framework is the

adaptation when it works at all. Prefer did or synthetic-control for policy-date designs.

- Extrapolation beyond the cutoff LATE needs added assumptions: multi-cutoff variation,

pre-period outcomes, ignorability-based, or derivative-based local extrapolation. Say which

when claiming anything away from the cutoff.

## R implementation

The complete runnable pipeline is scripts/rdd_template.R (estimation in both frameworks, fuzzy

diagnostics, the full falsification battery, power/MDE), with every call verified against the

rdpackages suite. The core:

```r

library(rdrobust); library(rddensity); library(rdlocrand)

rdplot(y, x, c = cutoff) # anatomy first

summary(rdrobust(y, x, c = cutoff)) # sharp: local linear, triangular, MSE-h, RBC

summary(rdrobust(d, x, c = cutoff)) # first stage / ITT on take-up

summary(rdrobust(y, x, c = cutoff, fuzzy = d))# fuzzy ratio

summary(rddensity(x, c = cutoff)) # manipulation

w <- rdwinselect(x, Z, c = cutoff) # local-randomization window

rdrandinf(y, x, cutoff = cutoff, wl = w$w_left, wr = w$w_right)

```

Package index with versions and links in references/details.md. Stata and Python mirrors of the

whole suite live at rdpackages.github.io; the guide ships full replication code in all three.

## Methods paragraph template

> Treatment assignment changes discontinuously at [cutoff] in [score], a rule set by

> [institution] before the outcomes we study, and units [cannot / can only imprecisely] control

> their score near it. We estimate the RD effect with local linear regression, a triangular

> kernel, and an MSE-optimal bandwidth, and report robust bias-corrected confidence intervals

> (Calonico, Cattaneo, and Titiunik 2014; Cattaneo and Titiunik 2022). We validate the design

> with the Cattaneo-Jansson-Ma density test and an exact binomial test, covariate balance at the

> cutoff with per-covariate bandwidths, placebo cutoffs, donut-hole estimates, and bandwidth

> sensitivity [and, for the fuzzy design, verify first-stage strength within the estimation

> bandwidth and report intention-to-treat effects alongside the complier estimate]. The estimate

> is local to the cutoff; a limitation of this design is that it does not identify effects for

> [units far from the threshold], and I extrapolate only [not at all / under the stated

> assumption], which costs [generalizability across the score distribution].

Every claim traces to references/canon.md; keys live in causal-design/references/causal.bib.

## Handoffs

- causal-design: whether an RD exists at all; where to go when the design gate fails.

- iv: the fuzzy branch's in-bandwidth first stage and exclusion argument live here;

weak-instrument inference (the F ladder, AR/CLR intervals) lives in iv, along with

many-instrument and shift-share logic.

- did / synthetic-control: policy-date designs masquerading as RD in time.

- preregister: pre-specifying an RD on an upcoming threshold change.