Regression Discontinuity

## The sharp design, precisely Write the running variable as `X` (the entrance exam score), the cutoff as `c = 60`, and the treatment indicator as `D`. In a **sharp** design, `D = 1 if X >= 60, and D = 0 if X < 60` with no exceptions. Every applicant scoring 60 or more receives the merit scholarship; nobody scoring 59 receives it. Treatment is a deterministic function of the score. Each applicant has two potential outcomes: `Y(1)`, the graduation outcome they would have with the scholarship, and `Y(0)`, the outcome they would have without it. Only one is ever observed, which is the fundamental problem the design has to work around. ## The identifying assumption The assumption is **continuity of the conditional mean potential outcomes at the cutoff**: `E[Y(1) | X = x]` and `E[Y(0) | X = x]` are both continuous in `x` at `x = 60`. Read that in plain words. Among applicants who *would not* get the scholarship, average graduation is a smooth function of exam score — a 59.9 scorer and a 60.1 scorer would be nearly identical in the untreated world. The same holds in the treated world. There is nothing magical about the number 60 itself, other than the scholarship rule attached to it. Notice what this does *not* say. It does not say the score is unrelated to graduation; higher scorers almost certainly graduate more, and the fitted trend on each side absorbs exactly that. It does not require randomisation. It does not require the two curves to be linear, or parallel, or anything else about their shape away from the cutoff. ## What follows: the estimand Under continuity, `lim(x -> 60 from above) E[Y | X = x] - lim(x -> 60 from below) E[Y | X = x] = E[Y(1) - Y(0) | X = 60]` The observed jump equals the average treatment effect **for units at the cutoff**. This is the whole payoff, and it is the sentence to be able to say precisely. The estimand is local in a specific sense: it describes applicants whose score is 60, the marginal ones. It does not describe the applicant who scored 90 and would have got the scholarship under any plausible rule, nor the applicant who scored 30. This is not a flaw to apologise for — it is often the exact policy quantity, because the live decision is usually whether to move the line to 58 or 62, and the marginal applicant is the one that decision is about. ## Estimation in practice Estimation means approximating those two limits. Fit a regression of the outcome on the score using only observations within a window around 60, separately on each side, and read the gap at the cutoff. A common specification pools the two sides: `Y = a + b*(X - 60) + t*D + g*D*(X - 60) + error`, fitted for `|X - 60| <= h` Here `t` is the estimated jump at the cutoff, the coefficient of interest, and centring the score at 60 is what makes `t` read off at the threshold rather than at score zero. Allowing separate slopes on each side (`g`) matters: forcing one common slope pushes curvature into the jump. ## Why continuity cannot be tested, and what you check instead Continuity is a statement about counterfactual averages, one of which is never observed on either side. You cannot test it. You can test its implications: - **Covariate smoothness.** Characteristics fixed before the exam — prior schooling, age, region — should not jump at 60. If they do, applicants on the two sides differ for reasons other than the scholarship, and the continuity story is in trouble. - **Density of the running variable.** If applicants can nudge their own score, the number of applicants just above 60 will be inflated relative to just below. - **Placebo cutoffs.** Re-run the estimate at fake thresholds like 50 or 70, where no rule exists. Repeated "significant" jumps there mean your specification manufactures jumps. - **A picture.** Plot binned outcome means against score with the fitted curves. If the jump is not visible to the eye at a sensible bin width, be sceptical of a jump that only a regression finds. ## Common mistakes Saying "RD gives you the average treatment effect" without the qualifier is the single most common error, and it is the one interviewers listen for. A second is describing continuity as "units near the cutoff are randomly assigned" — a useful intuition, but strictly it is a stronger claim than continuity, and stating it as the assumption suggests the definition was memorised rather than understood. A third is treating the running variable as a confounder to control away with a covariate adjustment, which destroys the design: the score is the axis of the design, not a nuisance. ## Reporting Report the estimated jump with its standard error, the window used, the specification, the picture, and the checks above. Say explicitly, in words, which units the number describes: applicants scoring around 60.