Paper Citation Record · LEDGER

Placebo Discontinuity Design

As of 24 August 2026, this Paper Citation Record lists 35 of 35 outbound references and 0 inbound Pith citation observations for arXiv:2507.12693.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2507.12693 v2

Coverage vector

measured 35 of 35 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T16:49:53.329406Z

measured 35 of 35 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-23T06:30:58.430688+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

35 of 35 outbound references displayed

verified exact0
verified fuzzy20
unresolved14
parse uncertain0
malformed identifier1
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation cfe3182c-4060-4902-822a-a211700f1d83 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:50:00.516453Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation ae194c63-e504-4891-892b-b0bee420ac3f · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:50:00.301890Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation cc0357e6-1965-4fdb-83f8-2d5bdedc79e8 · outbound

This paper cites 34 where the first line follows from Assumption 2d (exclusion) and the second line follows from Assumption 3b (𝜂𝑎⊥ ⊥𝑈|𝐷).

Placebo Discontinuity Design 34 where the first line follows from Assumption 2d (exclusion) and the second line follows from Assumption 3b (𝜂𝑎⊥ ⊥𝑈|𝐷)

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:50:00.079418Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 113d115d-a995-40a5-8045-cc5090fa966b · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:59.826537Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 2ab133e0-d811-4862-ad6c-5f734923ead8 · outbound

This paper cites The factoring of𝐴(𝐷,𝜂 𝑎)in (II) follows from Assumption 3 (𝜂𝑎 ⊥ ⊥𝜂𝑦,𝑈|𝐷=𝑑) and weak union, whereby𝜂 𝑎 ⊥ ⊥𝜂 𝑦|𝐷=𝑑,𝑈.

Placebo Discontinuity Design The factoring of𝐴(𝐷,𝜂 𝑎)in (II) follows from Assumption 3 (𝜂𝑎 ⊥ ⊥𝜂𝑦,𝑈|𝐷=𝑑) and weak union, whereby𝜂 𝑎 ⊥ ⊥𝜂 𝑦|𝐷=𝑑,𝑈

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:59.655530Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 3414a88e-c86e-4f56-baeb-e1b29ce14106 · outbound

This paper cites Specifically, to go from the second line to the third line, the initial terms cancel, and the later terms factorize.

Placebo Discontinuity Design Specifically, to go from the second line to the third line, the initial terms cancel, and the later terms factorize

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:59.424337Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 6411a1f5-2d6c-4406-9ff9-d45a8fa294ad · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 7

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:59.223694Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 7227d2ce-7385-4167-bd77-eee7f61c8c7d · outbound

This paper cites The limits on the right hand side exist due to Assumption 1 (RDD).

Placebo Discontinuity Design The limits on the right hand side exist due to Assumption 1 (RDD)

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:59.055185Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation e21ce0be-4924-4f44-a6a4-67524ffea099 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:58.880737Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 2c31eeba-3cec-47f0-9a86-0b1fc4573689 · outbound

This paper cites To begin, note thatE[𝑌|𝑑,𝑈 ] is unique.

Placebo Discontinuity Design To begin, note thatE[𝑌|𝑑,𝑈 ] is unique

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:58.696772Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 8fe3ddf1-ff2d-4b93-91d1-d44dace07b75 · outbound

This paper cites 42 The limits on the right hand side exist due to Assumption 1 (RDD).

Placebo Discontinuity Design 42 The limits on the right hand side exist due to Assumption 1 (RDD)

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:58.489916Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation fbdd501c-35c4-4e0e-9c89-439fa5b10bf5 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:58.310606Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 32042c45-42c7-47ff-9f65-dc8dc23056eb · outbound

This paper cites To begin, note thatE[𝑌|𝑑,𝑈 ] is unique.

Placebo Discontinuity Design To begin, note thatE[𝑌|𝑑,𝑈 ] is unique

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:58.073475Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation ddb6f8d3-4dbc-43ea-add0-40becc18e0c2 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 14

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:57.862477Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation e73e18dc-ded5-40d0-b4fa-92e984a71a4c · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:57.632413Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation b75ba303-7aff-412e-8f0b-9e3d798f37aa · outbound

This paper cites Specifically, to go from the second line to the third line, the initial terms cancel, and the later terms factorize.

Placebo Discontinuity Design Specifically, to go from the second line to the third line, the initial terms cancel, and the later terms factorize

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:57.402864Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 6c20e6b6-b2a0-4972-b7e1-e0f17fb2d1ae · outbound

This paper cites Rearranging and gives 𝜏0= E lim𝜖↓0[E[𝑌|𝐷=𝑑 ∗+𝜖,𝑈 ]−E[𝑌|𝐷=𝑑 ∗−𝜖,𝑈 ]]|𝐷=𝑑 ∗ E lim𝜖↓0[E[𝐴|𝐷=𝑑 ∗+𝜖,𝑈 ]−E[𝐴|𝐷=𝑑 ∗−𝜖,𝑈 ]]|𝐷=𝑑 ∗.

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:57.188900Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 1ce8dcb7-c114-48a9-b169-5ea17e47129c · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:57.036323Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation a23a33bb-db5b-4613-8961-34753ee2db9f · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:56.836289Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 344701a6-a533-4fa4-98f6-6bc9c793de72 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 20

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:56.622236Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 6471b702-c2f3-43c3-8556-f75601491142 · outbound

This paper cites 𝑛∑︁ 𝑖=1 𝜔𝑖,−𝑍𝑖 𝑊⊤ 𝑖 −𝑅⊤ 𝑖,1(R⊤ 1 K−R1)−1R⊤ 1 K−W #−1.

Placebo Discontinuity Design 𝑛∑︁ 𝑖=1 𝜔𝑖,−𝑍𝑖 𝑊⊤ 𝑖 −𝑅⊤ 𝑖,1(R⊤ 1 K−R1)−1R⊤ 1 K−W #−1

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:56.406241Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 0799055a-df1b-4f7f-b2db-505efb48c5e3 · outbound

This paper cites Standard results from e.g.

Placebo Discontinuity Design Standard results from e.g

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:56.257144Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation c43e94e2-a4a3-4d73-8140-d2e6dfec2222 · outbound

This paper cites Using Lemma E.1 (Q−-limit) and the continuous mapping theorem, we can write that, 𝑛Q−1 − 𝑝 →𝜅−1 lim 𝜖↓0 Cov𝑍𝑖,𝑊⊤ 𝑖 |𝐷=𝑑 ∗−𝜖 −1, 𝜅=𝑓 𝑑,+(𝑑∗) ∫ ∞ 0 𝐾(𝑢)d𝑢.

Placebo Discontinuity Design Using Lemma E.1 (Q−-limit) and the continuous mapping theorem, we can write that, 𝑛Q−1 − 𝑝 →𝜅−1 lim 𝜖↓0 Cov𝑍𝑖,𝑊⊤ 𝑖 |𝐷=𝑑 ∗−𝜖 −1, 𝜅=𝑓 𝑑,+(𝑑∗) ∫ ∞ 0 𝐾(𝑢)d𝑢

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:56.045940Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 59fa3022-05ad-47df-8bc4-bcc6fb86ba9d · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:55.862544Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 01d19f8d-8f74-4bc0-8ddd-caaba931948c · outbound

This paper cites Substituting inℎ+(𝑑−𝑑 ∗,𝑤)from Equation (1), lim 𝜖↓0 E[ℎ+(𝜖,𝑊)|𝐷=𝑑 ∗]=𝑔+(0)+E 𝑊⊤|𝐷=𝑑 ∗ 𝛾+ =𝛼+,0+ 𝛽𝑤 +,0 ⊤ 𝛾+.

Placebo Discontinuity Design Substituting inℎ+(𝑑−𝑑 ∗,𝑤)from Equation (1), lim 𝜖↓0 E[ℎ+(𝜖,𝑊)|𝐷=𝑑 ∗]=𝑔+(0)+E 𝑊⊤|𝐷=𝑑 ∗ 𝛾+ =𝛼+,0+ 𝛽𝑤 +,0 ⊤ 𝛾+

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:55.667970Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 21daab0e-90c9-453c-b208-da26b208819c · outbound

This paper cites By Lemma 3 (factuals) and Equation (1), for all𝑑∈D+(𝜖)recall that we can write, E[𝑌−ℎ +(𝐷−𝑑 ∗,𝑊)|𝐷=𝑑,𝑍=𝑧 ]=E 𝑌−𝑔 +(𝐷𝑖−𝑑∗)−𝑊 ⊤𝛾+|𝐷=𝑑,𝑍=𝑧 =0.

Placebo Discontinuity Design By Lemma 3 (factuals) and Equation (1), for all𝑑∈D+(𝜖)recall that we can write, E[𝑌−ℎ +(𝐷−𝑑 ∗,𝑊)|𝐷=𝑑,𝑍=𝑧 ]=E 𝑌−𝑔 +(𝐷𝑖−𝑑∗)−𝑊 ⊤𝛾+|𝐷=𝑑,𝑍=𝑧 =0

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:55.354695Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 794a0c8c-f78d-4768-92f8-1aa58787cad4 · outbound

This paper cites Applying the conditional expectationE[·|𝑍,𝐷=𝑑 ∗+𝜖] on both sides, E[𝑌|𝑍,𝐷=𝑑 ∗+𝜖]=𝛼+,0+𝛼+,1𝜖+E 𝑊⊤|𝑍,𝐷=𝑑 ∗+𝜖 𝛾++ 1 2𝑔(2) + (𝛿)𝜖 2.

Placebo Discontinuity Design Applying the conditional expectationE[·|𝑍,𝐷=𝑑 ∗+𝜖] on both sides, E[𝑌|𝑍,𝐷=𝑑 ∗+𝜖]=𝛼+,0+𝛼+,1𝜖+E 𝑊⊤|𝑍,𝐷=𝑑 ∗+𝜖 𝛾++ 1 2𝑔(2) + (𝛿)𝜖 2

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:55.106959Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 738ad12b-2705-4854-ab42-94d5cd283221 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 28

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:54.924298Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 61f84577-3378-4c0b-bfe8-2de92d231de2 · outbound

This paper cites In the proof of Proposition 1 (estimator decomposition), we showed that ˆ𝛼+,0 = ˆ𝛽𝑦 +,0−( ˆ𝛽𝑤 +,0)⊤ ˆ𝛾+.

Placebo Discontinuity Design In the proof of Proposition 1 (estimator decomposition), we showed that ˆ𝛼+,0 = ˆ𝛽𝑦 +,0−( ˆ𝛽𝑤 +,0)⊤ ˆ𝛾+

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:54.699228Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation c751e0c2-64d0-4e79-8952-a88f612217dc · outbound

This paper cites □ E.2 Technical lemma Lemma E.1(Q −-limit).Let Assumptions 7a-c and 8 hold, and assume thatℎ→0and 𝑛ℎ→∞.

Placebo Discontinuity Design □ E.2 Technical lemma Lemma E.1(Q −-limit).Let Assumptions 7a-c and 8 hold, and assume thatℎ→0and 𝑛ℎ→∞

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:54.412748Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 34394f40-7d3f-40c5-babb-96a3aec8bb69 · outbound

This paper cites an unresolved cited work.

Placebo Discontinuity Design Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:49:54.223402Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation a21de34f-aad4-4a70-9bd6-c62cfde67aa1 · outbound

This paper cites (2001, Section 4.1) imply that e⊤ 0(R⊤ 1 K−R1)−1R⊤ 1 K−W 𝑝 →(𝛽 𝑤 −,0)⊤=lim 𝜖↓0 E 𝑊⊤ 𝑖 |𝐷 𝑖 =𝑑∗−𝜖 54 1 ℎ𝑛 e⊤ 1(R⊤ 1 K−R1)−1R⊤ 1 K−W 𝑝 →(𝛽 𝑤 −,1)⊤.

Placebo Discontinuity Design (2001, Section 4.1) imply that e⊤ 0(R⊤ 1 K−R1)−1R⊤ 1 K−W 𝑝 →(𝛽 𝑤 −,0)⊤=lim 𝜖↓0 E 𝑊⊤ 𝑖 |𝐷 𝑖 =𝑑∗−𝜖 54 1 ℎ𝑛 e⊤ 1(R⊤ 1 K−R1)−1R⊤ 1 K−W 𝑝 →(𝛽 𝑤 −,1)⊤

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:54.030165Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 05824b0a-1945-4e0a-af80-2a33dcec5613 · outbound

This paper cites SetX 𝑖 = 1 ℎ𝑛 1 (𝐷𝑖 <𝑑∗)𝐾 𝐷𝑖−𝑑∗ ℎ𝑛 𝑍𝑖(𝑊𝑖−𝛽 𝑤 −,0)⊤ ∈R 𝑞×𝑞.

Placebo Discontinuity Design SetX 𝑖 = 1 ℎ𝑛 1 (𝐷𝑖 <𝑑∗)𝐾 𝐷𝑖−𝑑∗ ℎ𝑛 𝑍𝑖(𝑊𝑖−𝛽 𝑤 −,0)⊤ ∈R 𝑞×𝑞

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:53.826337Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 9f51f377-43c4-4254-9100-d0af1f8c1cbb · outbound

This paper cites We will show that 1 𝑛 𝑛∑︁ 𝑖=1 𝜔𝑖,−𝑍𝑖 𝑊⊤ 𝑖 −e⊤ 0(R⊤ 1 K−R1)−1R⊤ 1 K−W 𝑝 →𝜅lim 𝜖↓0 Cov𝑍𝑖,𝑊⊤ 𝑖 |𝐷 𝑖 =𝑑∗−𝜖.

Placebo Discontinuity Design We will show that 1 𝑛 𝑛∑︁ 𝑖=1 𝜔𝑖,−𝑍𝑖 𝑊⊤ 𝑖 −e⊤ 0(R⊤ 1 K−R1)−1R⊤ 1 K−W 𝑝 →𝜅lim 𝜖↓0 Cov𝑍𝑖,𝑊⊤ 𝑖 |𝐷 𝑖 =𝑑∗−𝜖

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:49:53.666154Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation f2eba07b-1d21-4906-aaee-6d91b75d2845 · outbound

This paper cites −”analogues of every object below are defined by replacing each“+.

Placebo Discontinuity Design −”analogues of every object below are defined by replacing each“+

Reference 35

Resolution: malformed identifier; raw_fallback, observed 2026-08-06T16:49:53.503120Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Pith citing papers

No inbound Pith citation observations are available.