Regression Discontinuity
Regression Discontinuity
The cutoff creates the comparison
Northbridge awards an offer when an application score reaches 70. Compare applicants just below and just above 70:
| Score | Offer rule | Completion rate |
|---|---|---|
| 68–69 | no offer | 0.49 |
| 70–71 | offer | 0.57 |
The eight-point gap is credible only if, without the offer, expected completion would change smoothly through 70. RDD does not claim that students with scores 40 and 90 are comparable.
For a sharp design with running variable , cutoff and treatment ,
This is a local effect at the threshold.
Draw before estimating
- plot the raw outcome against the running variable;
- show binned means only as a visual summary;
- fit separate local lines on each side;
- report the bandwidth and polynomial order;
- keep the cutoff and data support visible.
A global fifth-order polynomial may fit the sample yet invent extreme curvature near the cutoff. Local linear or local quadratic fits are usually easier to defend. Cattaneo, Idrobo and Titiunik (2020) provide a design-based practical treatment of estimation and diagnostics.
Bandwidth is a bias–variance decision
Suppose the estimated completion effect is:
| Window around 70 | Applicants | Estimate | Robust 95% interval |
|---|---|---|---|
| ±2 | 260 | 0.09 | −0.01, 0.19 |
| ±5 | 610 | 0.07 | 0.01, 0.13 |
| ±10 | 1,180 | 0.12 | 0.08, 0.16 |
The widest window is most precise but asks continuity over a larger range. Do not select the window because it gives the preferred -value. Use a principled bandwidth, then show nearby choices as sensitivity checks.
Sharp, fuzzy and kink designs
| Design | What changes at | Typical estimand |
|---|---|---|
| sharp RDD | treatment status from 0 to 1 | local treatment effect |
| fuzzy RDD | treatment probability jumps | local IV effect for cutoff compliers |
| regression kink | slope of treatment intensity changes | local effect per unit of intensity |
If offer probability rises from 0.10 to 0.75 at 70 and completion rises from 0.49 to 0.57, the fuzzy Wald estimate is
Interpret 12.3 percentage points as a local effect for applicants whose receipt is changed by crossing the cutoff—not an effect for every applicant.
Four threats and their probes
| Threat | Probe | What failure means |
|---|---|---|
| precise score manipulation | density and institutional audit | units around may not be comparable |
| another rule at 70 | policy inventory | discontinuity may combine treatments |
| predetermined covariate jump | balance plots and placebo outcomes | continuity is doubtful or coding is wrong |
| functional-form dependence | bandwidth/order sensitivity | estimate relies on extrapolation |
A density test is not a ritual. If assessors can round 69.8 to 70, inspect score production, heaping and appeals. Conversely, a visible density change does not identify its cause without institutional evidence.
What RDD does not identify automatically
- effects far from the cutoff;
- effects under a different treatment version;
- effects when units influence one another across the threshold;
- a policy expansion that changes capacity or applicant behaviour.
The rdrobust project implements local-polynomial estimation, robust bias correction and companion diagnostics. Software cannot establish that no other rule changes at the cutoff.
Quick check
Scores are recorded only as integers, and 35% of applicants score exactly 70. Is “use a narrower bandwidth” an adequate solution?