Regression Discontinuity and Synthetic Control

Both of these designs answer the same question, where does the counterfactual come from, with an unusually concrete answer. RD finds it just on the other side of an arbitrary cutoff. Synthetic control builds it from a weighted average of untreated units.

Regression discontinuity

Whenever treatment is assigned by a threshold on a continuous variable, a fraud score above 0.8, a customer above a spend tier, a student above a scholarship cut-off, units immediately on either side are nearly identical in everything except treatment. The threshold is arbitrary at the margin, so local randomisation is a defensible assumption rather than an assertion about the whole population.

The sharp RD estimand is the limit difference at the cutoff \(c\):

Estimation is local linear regression on each side within a bandwidth, and the bandwidth is the central bias-variance choice: wide uses more data and imports curvature that fakes a jump, narrow is unbiased and noisy. Data-driven bandwidth selection is standard practice rather than an option (Calonico, Cattaneo and Titiunik, 2014, Econometrica 82(6)).

Fuzzy RD applies when crossing the threshold changes treatment probability without determining it. The design becomes IV with the threshold indicator as instrument, and inherits LATE's interpretation.

The design's own diagnostic is manipulation. If units can precisely control their position relative to the cutoff, applicants who know the score and can adjust it, the local randomisation fails. A McCrary density test looks for a discontinuity in the density of the running variable at the cutoff, and a spike just above it is evidence of sorting. Continuity of pre-determined covariates across the cutoff is the second standard check.

Synthetic control

When one unit is treated and several are not, and no single control resembles the treated unit, construct one. Synthetic control chooses non-negative weights summing to one over the donor pool so the weighted combination matches the treated unit's pre-treatment outcome path and covariates, then uses that combination's post-treatment path as the counterfactual (Abadie, Diamond and Hainmueller, 2010, JASA 105(490), 493-505).

The California Proposition 99 study is the reference application: a synthetic California built from other states tracks actual California closely before 1988 and diverges after, with the estimate that by 2000 annual per-capita cigarette sales were about 26 packs lower than the synthetic counterfactual.

Two design properties are worth naming. The weights are transparent, so the counterfactual can be inspected and argued about, which a regression coefficient cannot. And the constraint that weights are non-negative and sum to one prevents extrapolation: the synthetic unit lies inside the convex hull of the donors, so if the treated unit is extreme on some dimension, a good match is impossible and the method says so rather than extrapolating silently.

Inference is by placebo permutation. Apply the method to each untreated unit in turn, treating it as if it had been treated, and compare the actual gap to the distribution of placebo gaps. With 38 donors, a treated gap larger than every placebo corresponds to roughly \(p = 1/39\).

When it breaks

RD is local, and that is a strong limit. The estimate applies at the cutoff. A scholarship's effect on students right at the threshold says little about students far above or below it, and extrapolating the RD estimate to the whole population is the most common misuse.

RD needs density at the cutoff. Power depends on observations near the threshold, not total sample. A million rows with a hundred near the cutoff is a small experiment.

Synthetic control needs a long, stable pre-period. Matching on a few pre-periods is easy and meaningless, since many weight vectors fit a short path. A good pre-treatment fit over many periods is what makes the counterfactual credible, and it is also why the method suits annual panel data more than short product time series.

Donor pool contamination. If donors are affected by the treatment, through trade, migration, or spillover, the synthetic counterfactual moves with the treatment and the estimate shrinks toward zero. Donors also must not have received similar interventions of their own.

Both designs measure one thing well and nothing else. RD gives a precise answer at a threshold; synthetic control gives one comparison for one unit. Neither generalises without an additional argument, and their credibility comes precisely from how narrow the claim is.