SDID — Synthetic Difference-in-Differences (Arkhangelsky et al. 2021)

SDID — Synthetic Difference-in-Differences (Arkhangelsky et al. 2021)#

Estimator:

Synthetic Difference-in-Differences (SDID) — mlsynth.SDID

Source:

Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021), “Synthetic Difference-in-Differences,” American Economic Review 111(12):4088-4118.

Replication type:

Path A — the paper’s Proposition 99 empirical, with a cross-validation against the authors’ own synthdid R package.

Status:

Fully verified — empirical headline reproduced and matched to the authors’ reference implementation.

Validation strategy#

Arkhangelsky et al.’s headline application is California’s Proposition 99 tobacco-control program, estimated on the canonical Abadie-Diamond-Hainmueller smoking panel (39 states, 1970-2000; California treated from 1989). The paper reports an SDID ATT of about -15.6 packs per capita, matched by the authors’ R synthdid package (-15.604). mlsynth reproduces that number to three significant figures, and we cross-validate the implementation against a live run of synthdid on the same matrix.

Path A — Proposition 99#

The panel ships as basedata/smoking_data.csv with a ready-made Proposition 99 indicator flagging the treated unit/period cells.

import pandas as pd
from mlsynth import SDID

df = pd.read_csv("basedata/smoking_data.csv")
df["treat"] = df["Proposition 99"].astype(int)
res = SDID({"df": df[["state", "year", "cigsale", "treat"]],
            "outcome": "cigsale", "treat": "treat",
            "unitid": "state", "time": "year",
            "display_graphs": False}).fit()
res.att                  # -15.605

mlsynth returns \(\widehat{\mathrm{ATT}} = -15.605\), matching the AER headline (-15.6) and the synthdid value (-15.604).

Cross-validation against the authors’ synthdid R#

We cross-validate against the method’s own authors’ code – the synth-inference/synthdid R package – run live on the identical outcome matrix. On Proposition 99 synthdid_estimate returns \(-15.6038\), and mlsynth reproduces it to \(1.6 \times 10^{-3}\) packs. The residual is the unit-weight ridge (\(\zeta\)) optimiser, not a methodological difference – the SDID weight QPs and the final regression are identical. The same package’s DiD (\(-27.349\)) and pure-SC (\(-19.620\)) estimates on the same matrix are recorded for context (mlsynth’s SDID targets the SDID column).

Quantity

mlsynth

synthdid R / AER

SDID ATT

-15.605

-15.604 / -15.6

DiD ATT (context)

—

-27.349

SC ATT (context)

—

-19.620

Durable check#

The reference is pinned under benchmarks/reference/sdid_prop99/ (R 4.3.3, synthdid commit 70c1ce3, data checksum), captured by its reference.R. The benchmark reads the frozen values (no R needed at test time):

python benchmarks/run_benchmarks.py --case sdid_prop99

It asserts the ATT lands on the published -15.604 (tol 0.05) and matches the authors’ synthdid to within \(0.02\) (observed \(1.6 \times 10^{-3}\)).

Synthetic triple difference — Virginia’s HPV mandate (Path A)#

The SC-DDD mode (subgroup / target_subgroup; Zhuang 2024) is cross-validated against the Stata sdid output of Feldman & Semprini (2026), who evaluate Virginia’s 2008 school-entry HPV vaccine mandate on cervical cancer incidence. Virginia is the treated state; ages 20-24 (the first mandate-exposed cohort by 2016) are the target subgroup; older age bands are the within-state controls. mlsynth demeans the age-adjusted incidence by the non-target ages within each treatment-group-by-year cell, then runs SDID on the 20-24 subgroup with Virginia treated from 2016.

The data ship as basedata/hpv_cervical_ddd.csv (39 states x 17 years, 2003-2019, public NPCR/SEER via the authors’ repository jsemprini/Virginia_HPVmandate_causal).

Estimator

mlsynth

Stata sdid

SC-DDD (transformed outcome)

+1.559

+1.559

naive SC-DD (untransformed 20-24)

+0.252

+0.252

The SC-DDD point estimate is a cell-for-cell match: mlsynth’s SDID engine is already validated against synthdid R above, so feeding it the Zhuang-demeaned outcome reproduces the Stata result exactly. The naive SC-DD on the untransformed 20-24 outcome lands on the paper’s near-null 0.252, so the triple-difference demeaning is what flips the estimate positive and significant. The placebo 95% interval excludes zero (mlsynth 0.35-2.76 vs the paper’s 0.42-2.70; the small difference is the placebo resampling, which is seed- and implementation- dependent). The durable case is benchmarks/cases/sdid_ddd_hpv.py:

python benchmarks/run_benchmarks.py --case sdid_ddd_hpv

References#

Feldman, C., & Semprini, J. (2026). “Causal inference, cancer registry data, and a single state policy change: Evaluating Virginia’s HPV vaccine mandate.” Journal of Cancer Policy 49:100777.

Zhuang, C. C. (2024). “A Way to Synthetic Triple Difference.” arXiv:2409.12353.

Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021). “Synthetic Difference-in-Differences.” American Economic Review 111(12):4088-4118.