What is the running variable in a regression discontinuity, and what makes age-65 Medicare eligibility a good one?
answer
- the score the rule is applied to
- a fixed threshold flips treatment on
- compare just below to just above
- can a person fake their score?
- beware other rules at the same value
basics
~20 sThe running variable is the score that decides treatment: everyone past a fixed cutoff gets it, everyone below does not. Age works well for Medicare eligibility at 65 because nobody can nudge their birth date across the threshold.
solid answer
~50 sA regression discontinuity design needs a numeric **running variable** and a **cutoff**: treatment is switched on by the rule "score at or above the cutoff". The design compares units just below the cutoff with units just above, on the argument that they are alike in everything except which side of the rule they landed on, so the jump in the outcome at the cutoff is attributable to the treatment. Age as a running variable for Medicare eligibility at 65 is strong because age is measured precisely, nobody can move themselves across the threshold, and someone 64 years and 11 months old is not systematically different from someone 65 years and 1 month old. The design still fails if some *other* rule switches on at the same age, because then two things jump at once and the estimate cannot separate them.
go deeper
Be ready to name the two objects out loud — running variable and cutoff — and to say that the comparison is between units just below and just above, not between all treated and all untreated.
You should be able to explain why a trend is fitted on each side of the cutoff and why the score being correlated with the outcome is fine, as long as that relationship does not break at the threshold.
Show judgment about which real business rules qualify: is the score self-reported, is the cutoff enforced mechanically, does anything else fire at the same value? Interviewers want the screening instinct, not the formula.
Own the framing question of when an eligibility rule is worth studying at all: the design answers about units at the line, so it pays off when the decision on the table is where to set that line.
## What the design is A regression discontinuity (RD) design exploits a rule of the form "you get the treatment if and only if your score reaches a threshold". Two objects define it: - The **running variable** (also called the forcing or assignment variable): the numeric score the rule is applied to. Exam score, age, income, points balance, calendar date. - The **cutoff**: the fixed value at which the rule flips. Score 60, age 65, 1,000 points. Treatment is a deterministic function of the running variable in the simplest case: units at or above the cutoff are treated, units below are not. ## Why a cutoff can identify a causal effect Comparing everyone treated with everyone untreated is hopeless here, because the two groups differ by construction — treated units have higher scores, and score is usually related to the outcome. RD sidesteps this by looking only *locally*. Just below and just above the threshold, units are nearly identical on the running variable itself, and there is no reason for anything else about them to change abruptly at that exact value. Any discrete jump in the average outcome at the cutoff is therefore attributed to treatment, while the smooth relationship between score and outcome on either side is subtracted away by fitting a trend on each side. Pictorially, you plot the average outcome against the running variable in bins, fit a curve to the left of the cutoff and a separate curve to the right, and read the vertical gap between the two fitted lines at the cutoff. ## What makes a good running variable Three properties matter. 1. **It is measured precisely and finely.** A score recorded in ten wide buckets gives you nothing to compare near the threshold. Age measured in days or exam score measured in points gives plenty of units on both sides. 2. **Units cannot precisely control where they land.** If a person can push their own score just over the line, then the units just above the cutoff are the ones who wanted the treatment and knew how to get it, which is exactly the selection RD is supposed to avoid. Date of birth is unfakeable; a self-reported income figure on a benefits form is not. 3. **Nothing else changes at the same value.** The design attributes the whole jump to the studied treatment, so it needs the cutoff to be special for one reason only. Age-65 Medicare eligibility scores well on the first two: age is exact, continuous in practice, and immune to manipulation. Point three is the honest caveat — 65 is a culturally loaded age for retirement, so a careful analyst has to argue that the jump they see is health-coverage eligibility rather than another rule or behavioural change hitting the same birthday. ## The local comparison, and its price Because the argument only holds near the cutoff, RD answers a narrow question: what does the treatment do to units sitting *at* the threshold? That is a genuine causal effect for a real, well-defined set of units, and it is often exactly the policy question ("should we move the eligibility line?"). It is not, by itself, the effect for units far from the cutoff, and a candidate who claims it is has misunderstood what the design bought. ## Terminology to keep straight - The **running variable** is not the outcome. The outcome is what you measure afterwards — hospital admissions, graduation, spend. - The running variable is not a nuisance covariate to be balanced away. It is the axis the whole design is built on, and it is deliberately *not* balanced across the two groups: units above the cutoff always have higher scores. That is fine, because the fitted trend on each side absorbs the smooth score–outcome relationship. - The cutoff must be a rule that was actually enforced. If eligibility was decided by a committee that "usually" follows the score, treatment does not switch cleanly at the threshold and you are in a different, fuzzier version of the design. ## How an interviewer probes this Expect to be handed a business rule — a discount unlocked at a spend level, a review triggered above a risk score — and asked whether it supports an RD, what the running variable and cutoff are, and what would break the argument. The strongest short answer names the two objects, states the local comparison, and immediately raises manipulability and co-occurring rules as the things you would check first.
- What kind of running variable would make you refuse to run the design?One the unit controls or reports itself, one recorded in coarse buckets so there is nothing to compare near the threshold, or one where the cutoff was applied loosely rather than mechanically. Self-reported income on an application form fails on the first count; a five-band satisfaction rating fails on the second.
- Why fit a separate trend on each side instead of just comparing two group means?Because the outcome usually varies smoothly with the score, so a raw mean difference mixes the treatment jump with that slope. Fitting a line on each side and reading the gap at the cutoff removes the smooth part and leaves the discrete jump, which is the quantity the design identifies.
- Does regression discontinuity need the running variable to be unrelated to the outcome?No, and it usually is related — higher scorers typically have better outcomes anyway. The design only needs that relationship to be smooth through the cutoff. What must not happen is a break in the score–outcome relationship at the threshold for reasons other than treatment.
It is like a height bar at a theme-park ride: the child a centimetre under and the child a centimetre over are the same child in every way that matters, but only one gets on.
saying these in an interview costs you the question
- Compares all treated units with all untreated units
- Calls the running variable a confounder to be balanced away
- Assumes any eligibility rule automatically yields a valid design
- Treats a self-reported score as unmanipulable
- Ignores other programmes switching on at the same cutoff