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

By Assumptions 1 (RDD), 4 (continuity), we can take the limit of the equality above to show, lim ๐œ–โ†“0 [E[๐‘Œ|๐ท=๐‘‘ โˆ—+๐œ–,๐‘ˆ ]โˆ’E[๐‘Œ|๐ท=๐‘‘ โˆ—โˆ’๐œ–,๐‘ˆ ]] =lim ๐œ–โ†“0 E ๐‘Œ(0,๐‘‘ โˆ—+๐œ–,๐‘ˆ,๐œ‚ ๐‘ฆ)|๐ท=๐‘‘ โˆ—+๐œ–,๐‘ˆ โˆ’E ๐‘Œ

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Placebo Discontinuity Design

econ.EM ยท 2025-07-16 ยท conditional ยท novelty 7.0

Even when strategic manipulation makes the running variable jump at the cutoff, the treatment effect can be recovered by subtracting the placebo outcome's discontinuity, weighted by a local instrumental variable estimate, from the usual RDD estimate.

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  • Placebo Discontinuity Design econ.EM ยท 2025-07-16 ยท conditional ยท none ยท ref 6

    Even when strategic manipulation makes the running variable jump at the cutoff, the treatment effect can be recovered by subtracting the placebo outcome's discontinuity, weighted by a local instrumental variable estimate, from the usual RDD estimate.