Regression discontinuity uses rules that switch treatment on at a cut-off to estimate causal effects.
Such rules are everywhere: exam pass marks, age limits, income thresholds and credit score cut-offs.
The design works because units just either side of a cut-off are nearly the same in every way except treatment.
The running variable is the score that decides treatment, such as an exam mark.
Outcomes are plotted against the running variable, and a line is fitted on each side of the cut-off.
The size of the jump between the 2 lines at the cut-off is the estimated effect.
A narrow window around the cut-off gives fairer comparisons but fewer data points.
A wide window gives more data but relies more on the fitted lines having the right shape.
Manipulation is the main threat, such as markers nudging scores of 69 up to 70.
A pile-up of units just above the cut-off in a histogram is the usual sign of manipulation.
Other characteristics, such as age or past marks, should not jump at the cut-off, which offers a check on the design.
In fuzzy designs the cut-off changes the chance of treatment rather than fixing it, and an instrumental variable estimate is used.
The estimate applies near the cut-off, so it says little about units far from it.
In practice it is among the most trusted designs without random assignment, because its assumptions can be checked visually.
The usual mistake is fitting curves that bend sharply near the cut-off, which can invent or hide a jump.