Regression discontinuity is one of the cleanest natural experiments in econometrics — and one an examiner loves, because a single picture carries the argument. It turns up in every second-year metrics course, New York’s included, and rewards students who can say what the jump at the cutoff does and does not measure. This page builds the sharp design from the running variable up.

1 · The idea — treatment assigned by a cutoff

A regression discontinuity design exploits a rule: some benefit is handed out whenever a score crosses a threshold — a scholarship above a mark, a smaller class below an enrolment cap, a warning past an attendance limit.

Call the score the running variable X and the threshold the cutoff c. In a sharp design the treatment D is a deterministic step at the cutoff:

D = 1 when X ≥ c, and D = 0 when X < c.

Everyone above the line is treated, everyone below is not.

2 · The estimand — the jump in the conditional mean

A student who scored just below the cutoff and one just above are, in everything you did not measure, near-identical — the half-mark between them is mostly luck. The cutoff thus splits that band into treated and untreated groups — comparing them is almost a randomised experiment.

The effect the design identifies is the jump in the conditional expectation of the outcome at the cutoff:

τ = (limit of E[Y | X] as X falls to c) − (limit as X rises to c).

The identifying assumption is continuity: absent the treatment, average outcomes would move smoothly through c, so any break in E[Y | X] at the cutoff is the treatment’s. Read τ carefully: it is the effect for students at the threshold, not for everyone.

3 · Fitting the two sides, and the bandwidth

You estimate τ by fitting the outcome against the running variable separately on each side of the cutoff, then taking the gap between the two fitted values at c. One line across the whole range would smear the jump into the slope.

Which observations count? Only those within a window of width h around the cutoff — the bandwidth — and the choice is a genuine trade-off:

• Narrow h: points close to c, where a line fits well, so bias is small — but few students, so variance is high.
• Wide h: more data lowers variance — but distant points, where the relationship curves, bend the line and bias grows.

You trade one error for the other; the worked example shows it biting.

4 · Checking the design holds

Continuity is an assumption, not a fact. Three checks defend it, each a standard exam point.

• Manipulation of the running variable. If students can precisely control their score — a grader nudging a 69 to a 70 for the scholarship — the just-above group is no longer comparable to the just-below. The McCrary density test looks for bunching: a spike in students landing just past the cutoff is the warning sign.
• Covariate smoothness. Characteristics fixed before treatment — prior grades, age, background — should not jump at the cutoff. If they do, the groups differ for reasons other than treatment.
• No other rule at the threshold. If a second programme switches on at the same cutoff, the jump you measure mixes both.

Two boundaries. When treatment only becomes more likely at the cutoff, the design is fuzzy, and the outcome jump is rescaled by the jump in treatment probability — an instrumental-variables reading this site treats under its own topic. And unlike difference-in-differences, which compares two groups trending in parallel over time, RD leans on nothing but continuity at a single point.

Worked example — a merit scholarship at a score cutoff

A university awards a merit scholarship to every applicant scoring X ≥ 70 on its entrance exam. The outcome Y is end-of-first-year percentage; you want the scholarship’s effect at the margin.

Step 1 — The design. Treatment is a sharp step: D = 1 for X ≥ 70, D = 0 below. The cutoff is c = 70.

Step 2 — Fit the control side. For students just below the cutoff (scores 60–70), OLS gives Ŷ₋ = 64 + 0.2(X − 70). At the cutoff its fitted value is 64.

Step 3 — Fit the treated side. For scholarship students just above (scores 70–80), OLS gives Ŷ₊ = 72 + 0.3(X − 70). At the cutoff its fitted value is 72.

Step 4 — The jump. Subtract the two fitted values at c: τ̂ = 72 − 64 = 8. The scholarship lifts the first-year average by about 8 points at the margin.

Step 5 — Widen the bandwidth. Now include weaker applicants down to score 55, whose mean outcomes are 58, 62, 63 at scores 55, 60, 65 — rising then flattening, so concave. Refit one line over the wider window and OLS returns Ŷ₋ = 66 + 0.5(X − 70): steeper slope, fitted value 66 at the cutoff.

Step 6 — See the bias. The low-scoring students sit below what the near-cutoff trend predicts, so a single line is dragged steeper and overshoots — 64 becomes 66. The jump shrinks to τ̂ = 72 − 66 = 6: two points of pure curvature.

Step 7 — Resolve. The narrow estimate (8) is nearer the truth, because a straight line fits well right at the cutoff; the wide window bought lower variance at the cost of bias. Report the jump at your chosen bandwidth, check it holds as you vary h, and remember it is the local effect at X = 70.

Can you say exactly what the jump at the cutoff measures — and what it doesn’t? That reading is what an RD answer is marked on: the effect at the threshold, identified by continuity, reported at a bandwidth you can defend. A one-on-one econometrics tutor drills the two-sided fit and the validity checks with you until they come out cleanly under exam pressure. Book a trial session.

Practice

Q1. With c = 80, the fitted lines are Ŷ₋ = 12 + 0.5(X − 80) below and Ŷ₊ = 17 + 0.3(X − 80) above. Compute the estimated jump τ̂.

Q2. Below-cutoff mean outcomes are 30, 34, 35 at scores 40, 45, 50; the treated-side fit predicts 43 at the cutoff c = 55. Estimate the jump (a) with a narrow bandwidth using the two bins nearest the cutoff, and (b) with a wide bandwidth using all three.

Q3. With c = 40, the fitted lines in raw form are Ŷ₋ = 20 + 0.3X below and Ŷ₊ = 12 + 0.6X above. Compute τ̂.

Answers. Q1: at X = 80 both centred terms vanish, so τ̂ = 17 − 12 = 5. Q2: (a) the line through (45, 34) and (50, 35) has slope 0.2 and reaches 36 at the cutoff, so τ̂ = 43 − 36 = 7; (b) OLS through all three points has slope 0.5 and reaches 38, so τ̂ = 43 − 38 = 5 — the wider window imports the tail’s curvature. Q3: Ŷ₊(40) = 12 + 24 = 36 and Ŷ₋(40) = 20 + 12 = 32, so τ̂ = 36 − 32 = 4.

Key takeaways

• A sharp design assigns treatment by a deterministic step at a cutoff on the running variable: below it untreated, at or above it treated.
• The estimand is the jump in the conditional mean at the cutoff — the effect for students at the threshold, identified by continuity.
• Fit separate lines each side and take the gap between the fitted values at the cutoff. Bandwidth trades bias (curvature, when wide) against variance (few points, when narrow).
• Defend the design: no manipulation of the running variable, smooth covariates through the cutoff, no second rule at the threshold.

Why New York students choose our econometrics tutoring

• PhD tutors in applied econometrics: sessions are led by tutors who estimate and diagnose these designs in their own research, not by generalists a chapter ahead.
• Matched to your course: the notation, bandwidth conventions and software commands come from your own module and problem sets.
• Online and one-on-one: each session runs over a shared screen on your past papers, so the work lands on the questions your exam will ask.

FAQ

Q: What is a regression discontinuity design?
A: A way to measure a causal effect when treatment is set by a strict cutoff rule. You compare outcomes just above and just below the threshold, where students are otherwise alike, and read the effect off the jump in the outcome at the cutoff.

Q: What is the running variable?
A: The score the cutoff is applied to — an exam mark, an income level, a date. Treatment depends on whether it clears the threshold, so it sits on the horizontal axis of every RD plot.

Q: What is the difference between a sharp and a fuzzy design?
A: In a sharp design, crossing the cutoff sets treatment exactly. In a fuzzy design it only changes the probability of treatment, so the outcome jump is rescaled by the jump in take-up. This page covers the sharp case.

Q: How do you choose the bandwidth?
A: Balance bias against variance: a narrow window lowers bias but raises variance, a wide one does the reverse. Report the estimate across several bandwidths and check it holds steady.

Q: How do you know an RD design is valid?
A: Test its assumptions — no precise manipulation of the running variable, no jump in pre-treatment characteristics at the cutoff, and no other programme sharing the threshold. Failing any one undermines the comparison.

Book an econometrics tutor in New York for regression discontinuity — and the rest of the course

The examiner’s marks sit on the details: what τ measures, why the bandwidth matters, which validity check catches which failure. One-on-one online sessions drill those on your own module’s past papers. Tell your tutor your course and exam date, and get matched this week.