Replications

Every estimator in mlsynth is implemented from a specific published source paper, and most of them carry an explicit replication section in their documentation – either reproducing an empirical result on the original authors’ dataset (“Path A”), reproducing a Monte Carlo from the paper’s simulation section (“Path B”), or matching the output of an authoritative reference implementation (“cross-validation”).

This page catalogues those replications. Thirty-two of the thirty-six estimators are currently fully verified, three carry partial replications (one-draw illustrations or empirical applications without an explicit number-match), and one has no replication content yet.

• The Canonical workhorses section covers the workhorse SC variants you would reach for first.

• The Decomposition-first synthetic control section covers decomposition-first methods that match on a cycle or spectral component, not raw outcomes.

• The Generalising the estimand, treatment, or unit section covers estimators that generalise the classical setup along one axis – a different estimand, treatment type, or unit scale.

• The Convex-hull relaxation section covers estimators that relax the convex-hull assumption.

• The High-dimensional donor pools section covers high-dimensional donor pools.

• The Time-aware and factor models section covers state-space and factor-model methods.

• The Staggered adoption, Missing data, and Identification under endogeneity sections cover the remaining observational families.

• The Experimental design section covers experimental design.

Each entry below links to the full replication code in the source documentation page and calls out the headline number that demonstrates the match.

Canonical workhorses

• Two-Step Synthetic Control – Li & Shankar (2024) Two-Step Synthetic Control. Path A: ATT (+1131.97) and pre-RMSE (434.43) on the Brooklyn-showroom panel reproduced to three decimals; Step 1’s recommendation matches the paper’s MSC(b). Path B: Figure 2 MSE-ratio grid – all 16 cells of the \((T_1, T_2)\) sweep fall below \(1.0\), range \([0.039, 0.889]\), matching the paper’s geometry.

• Forward Difference-in-Differences (FDID) – Li (2024) Forward Difference-in-Differences. Path B: Table 5 PMSE grid across DGP1-DGP4 at four \((T_1, T_2)\) configurations; cell \((48, 24)\) yields \(\mathrm{PMSE} = 0.084\) against the paper’s \(0.082\).

Decomposition-first synthetic control

• Harmonic Synthetic Control (HSC) – Harmonic Synthetic Control. Path A: the 1997 Hong Kong handover – 2003 effect \(-\$1{,}902\) against the paper’s \(-\$1{,}900\); Korea weight 0.18 and Germany weight 0.14 match the paper. Path B: Liu-Xu regime-adaptation grid with ARIMA(1,1,0) trends – HSC RMSE 6.46 vs. SC-diff 6.11 under the idiosyncratic regime.

• Synthetic Business Cycle (SBC) – Synthetic Business Cycle. Path A: German reunification – Greece weight 0.44, Netherlands 0.37, \(\widehat{\mathrm{ATT}} \approx -952\); also reproduces the Hong Kong handover. Path B: Shi-Xi-Xie Models 1-3 nonstationary DGP, Model 1 MSE ratio 0.012 vs. the paper’s 0.01.

Generalising the estimand, treatment, or unit

• Synthetic Control with Multiple Outcomes (SCMO) – Tian, Lee & Panchenko (2024) multi-outcome SC. Path A: German reunification (nine pre-1989 indicators) – pre-RMSE = 110, matching Table 1 to three decimals. Path B: TLP Table 1 plus SBMF averaging, 400-800 reps; bias / SD at \(K = 6\) are \(1.11 / 1.40\) against the paper’s \(1.08 / 1.36\).

• Continuous-Treatment Synthetic Control (CTSC) – Continuous-Treatment SC. Path B: Section 5 / Table 1 Monte Carlo across Models 1-4 with factor structure and true average effect \(= 0\); CTSC bias \(\approx 0.00\) against the paper’s \(0.005\)-\(0.011\) range, vs. fixed-effects bias \(\approx 0.80\).

• Distributional Synthetic Control (DSC) – Distributional SC. Path B: Gunsilius (2023) Figure 4 reproduction – the 2-Wasserstein-squared distance collapses at \(J = 30\) against the rough fit at \(J = 4\), and a location-shift planted effect of \(+1.5\) is recovered.

• Synthetic Interventions (SI) – Synthetic Interventions. Path A: California Proposition 99 multi-arm counterfactual reproduces the paper’s reported control/tax-only/full-program values 75.8 / 57.5 / 59.1. Path B: Gavish-Donoho \(k = 5\) for control and \(k = 1\) for the tax / program arms, matching the paper.

• MicroSynth (User-Level Balancing SC) – User-Level SC. Path B: Section 5 contamination-recovery study, 200 reps with true lift \(+0.05\); MicroSynth bias \(+0.0028\) vs. ITT bias \(-0.0181\) and naive TOT bias \(+0.0291\).

Convex-hull relaxation

• Imperfect Synthetic Controls (ISCM) – Imperfect Synthetic Controls. Example: one-factor panel with planted effect 3.0 recovered as \(\widehat{\mathrm{ATT}} \approx 3.0\) and a small outside- hull fit metric, illustrating the all-units-as-synthetic- controls construction. Full Monte Carlo replication queued.

• Nonlinear Synthetic Control (NSC) – Nonlinear SC. Path B (one-draw): Tian (2023) Section 4 nonlinear DGP with \(J = 12\), \(T = 18\), \(\tau_{\mathrm{true}} = 0.10\) – recovered \(\widehat{\tau} \approx 0.10\). Averaged Monte Carlo queued.

High-dimensional donor pools

• Bayesian Synthetic Control with a Soft Simplex Constraint (BVS-SS) – Xu & Zhou (2025) Bayesian SC with soft simplex. Path A: China anti-corruption study on luxury-watch imports (87 commodity-category donors) – ATT = -0.020 (95% credible interval \([-0.033, -0.003]\)) matches the paper’s \(-0.021\ ([-0.032, -0.008])\).

• Cluster Synthetic Controls (CLUSTERSC) – Cluster-then-pool SC. Path A (PCR-SC): California Proposition 99 – rank-1 PCR \(\widehat{\mathrm{ATT}}\) in the \([-19, -24]\) range matching the classical ADH baseline. Path A (RPCA-SC): West German reunification – Norway weight 0.48, France 0.35, pre-RMSE \(\approx 90\) matching Bayani’s reference figures. Path B: Amjad-Shah-Shen periodicity DGP plus a two-factor DGP for missing-data robustness.

• Multi-Level Synthetic Control (mlSC) – Bottmer (2024) multi-level SC. Path A: matched to the reference implementation on the README DGP – \(\widehat{\mathrm{ATT}} = +0.011930\), \(\lambda = 1.970185\), equal to six decimals. Path B: 200 independent draws, \(\max |\Delta| = 5.76 \times 10^{-4}\).

• Panel Data Approach (PDA) – Shi & Huang (2023) panel data approach. Path A: Hong Kong economic integration (HCW 24-economy panel, quarterly GDP growth). Path B: Table 1 forward-selection vs. LASSO at 300 reps – size = 0.047 vs. the paper’s \(\approx 0.05\); power = 0.98 at D5 vs. \(1.0\).

• Relaxed / Penalized Synthetic Control (RESCM) – Liao, Shi & Zheng (2025) relaxed SC. Path A: Proposition 99 (38 control states). Path B: high-dim grid \(N = 90\), \(T_1 = T_2 = 36\), 50 reps at \(\delta \in \{0, 1\}\); SC and LINF achieve size 0.20-0.22 and power 0.98; RELAX-\(L_2\) reaches size 0.28, power 0.94.

• Forward-Selected Synthetic Control (FSCM) – Cerulli (2024) forward-selected SC. Path A: Proposition 99 – \(\widehat{\mathrm{ATT}} = -20.15\), pre-\(R^2 = 0.970\), CV RMSPE \(= 1.605\) vs. the full-pool baseline 2.916.

• Sparse Synthetic Control (SparseSC) – Sparse SC. Path A: California Proposition 99 with the augmented ADH-7 dataset of Vives, with conformal inference – validation MSE conformal CI \(\approx [-20, -18]\), magnitude consistent with the ADH baseline.

Time-aware and factor models

• Factor Model Approach (FMA) – Li & Sonnier (2023) factor model approach. Path B: coverage study under DGP1 (stationary) and DGP2 (non-stationary), 1000 reps; coverage at \(M = 1000\) equals 0.947 against the nominal 0.95.

• Time-Aware Synthetic Control (TASC) – Rho et al. (2026) time-aware SC. Path A: Proposition 99 – pre-RMSE 0.767, ATT -16.793, gap of -24 packs by 2000 against the paper’s Figure 10 gap of -25 to -30. Path B: Section 5.2 \((Q, R)\) ablation grid (30 reps); TASC dominates the simplex-SC baseline in all four cells, with a \(4.5\times\) margin at big-\(Q\) / small-\(R\).

Staggered adoption

• Synthetic Difference-in-Differences (SDID) – Arkhangelsky et al. (2021) synthetic difference-in-differences. Path A: California Proposition 99 – \(\widehat{\mathrm{ATT}} = -15.605\) matches the paper’s \(-15.6\) (and the synthdid R-package’s \(-15.604\)) to three significant figures; placebo SE \(7.58\) against the paper’s \(8.4\).

• Spatial Synthetic Difference-in-Differences (SpSyDiD) – Spatial Synthetic-DiD. Path B: 8x8 grid with 6 treated units and planted spillover-exposure effects – \(\widehat{\tau} \approx 2.0\), \(\widehat{\tau}_s \approx 1.0\), both recovered.

• Spillover-Detecting Synthetic Control (SPOTSYNTH) – O’Riordan & Gilligan-Lee (2025) spillover detection for donor selection. Path B: the Figure 2 bias study on the Appendix B DGP – a synthetic control on all donors is biased (\(\approx +1.6\)), one on valid donors is unbiased, and the S1 / S2 screens recover most of the gap, degrading with noise. Path A (semi-synthetic): the Figure 6 demonstrations – a planted noisy-proxy donor that biases California tobacco (\(-1.4\) vs the canonical \(-20.5\)) and German reunification toward zero is flagged and excluded, restoring the effect; both run through SPOTSYNTH.fit().

• Partially Pooled SCM (PPSCM) – Ben-Michael, Feller & Rothstein (2022) partially pooled SC. Cross-validation: matched end-to-end to the augsynth R-package vignette (\(\nu = 0.2607\), average ATT \(= -0.011\)) to four decimals.

• Staggered Synthetic Control (SSC) – Cao, Lu & Wu (2026) staggered synthetic control. Path A: the Guanajuato police-reform application (Section 4; \(N = 33\), 10 staggered adopters) – event-time ATT estimates match the authors’ reference output for all seven outcomes, to \(\approx 10^{-4}\) for the homicide (\(T_0 = 174\)) and theft rates and \(\approx 10^{-3}\) for the annual cartel outcomes, with end-of-sample bands present/NaN exactly as in the reference. Path B: the paper’s staggered AR(1)-factor DGP (Section 3) – the event-time ATT path \(\tau = 1 + e\) is recovered, using all units (including not-yet-treated) as donors.

Missing data

• Matrix Completion with Nuclear Norm Minimization (MCNNM) – Athey, Bayati, Doudchenko, Imbens & Khosravi (2021) matrix-completion SC. Path A: Proposition 99 – \(\widehat{\mathrm{ATT}} \approx -20\) packs per capita, consistent with classical SC, FSCM and SNN estimates on the same panel.

• Synthetic Nearest Neighbors / Causal Matrix Completion (SNN) – Synthetic Nearest Neighbours. Cross-validation: Proposition 99 – \(\widehat{\mathrm{ATT}} \approx -19\) packs/capita, matching the Abadie-Diamond-Hainmueller (2010) reference baseline.

• Robust Matrix estimation with Side Information (RMSI) – Agarwal, Choi & Yuan (2026) robust matrix estimation with side information. Path A: Proposition 99 with the Abadie covariates as side information – \(\widehat{\mathrm{ATT}} \approx -21\) packs/capita (widening to \(\approx -32\) by 2000), matching the ADH baseline. Path B: the paper’s four-component MNAR Monte Carlo (Section 5.1) – RMSI’s missing-block AMSE is lower than the no-side-information baseline, the paper’s central finding.

Identification under endogeneity

• Synthetic IV – Gulek & Vives (2024) synthetic IV. Path A: Autor-Dorn-Hanson China-shock 2SLS (Table 3) reproduced exactly on the public replication archive – coefficients \(-0.888 / -0.718 / -0.746\) against \(-0.89 / -0.72 / -0.75\). Path B: Section 6 Syrian- calibrated Monte Carlo across \(r \in \{0.5, 0.7, 0.9\}\) at 200 reps; TSLS-TWFE bias 0.111 / 0.228 / 0.387 matches the paper’s 0.111 / 0.218 / 0.360 essentially exactly.

• Proximal Inference Synthetic Control (PROXIMAL) – Proximal SC. Path A: Panic of 1907 on the Trust panel, reproduced across all six proximal variants – PI \(-1.148\) vs. paper \(-1.138\); PIS \(-1.148\) vs. \(-1.134\); PIPost \(-1.220\) vs. \(-1.220\). Path B: Single-Proxy SC (SPSC) plus Doubly-Robust and PIPW Monte Carlo – SPSC-DT \(\widehat{\mathrm{ATT}} = -0.815\) against \(-0.816\).

Experimental design

• Synthetic Design (SYNDES) – Abadie & Zhao (2025) synthetic design. Path B: Section 5 ablation across the three design types (per_unit / two_way_global / one_way_global) at 40 reps under AR(1) factors; per_unit design RMSE = 0.098 against a random-DiM baseline of 0.982, power = 0.50 at MDE = 0.157.

• Synthetic Controls for Experimental Design (MAREX) – Abadie & Zhao (2025) market-exclusion design. Path A: Walmart 45-store weekly-sales placebo design – pre-fit RMSE 2.2%, placebo ATT \(-1.0%\), p-value 0.937 against the paper’s 0.933. Path B: Lin-factor DGP recovers the treatment effect across the MAE-vs-effect-scale sweep.

• Parallel-Trends Supergeo Design (PANGEO) – Parallel-Trends Supergeo Design. Path B: seasonal-sales panel with up-trend, down-trend, and cyclical trajectories; PANGEO RMSE \(\approx 0.2\) against scalar- matching RMSE \(\approx 6\) at \(\tau = 4\) (a \(30\times\) improvement); pre-period parallelism \(R^2 \in [0.90, 0.98]\).

• Synthetic Principal Component Design (SPCD) – Synthetic Principal Component Design. Path B: Lu-Li-Ying-Blanchet linear-factor model with \(\tau = 1\), \(\sigma = 1\) – SPCD mean \(\widehat{\mathrm{ATT}} \approx 1.0\), RMSE \(\approx 0.4\) against the random-design baseline RMSE \(\approx 3.9\) (a \(10\times\) improvement).

• Lexicographic Synthetic Control (LEXSCM) – Lexicographic SC. Path B: synthetic sales panel with budget and operational constraints – enumeration solver returns OPTIMAL status and best imbalance \(\approx 4 \times 10^{-4}\), demonstrating the full lexicographic max-validity / max-power decision.

Coverage summary

Verification coverage by family

Family	Verified	In family	Status
Canonical workhorses	2	2	Complete (TSSC, FDID)
Decomposition-first	2	2	Complete (HSC, SBC)
Generalised estimand / treatment / unit	5	5	Complete (SCMO, CTSC, DSC, SI, MicroSynth)
Convex-hull relaxation	0 (2 partial)	2	ISCM and NSC have one-draw illustrations; averaged Monte Carlos queued
High-dimensional donors	7	7	Complete (BVSS, CLUSTERSC, MLSC, PDA, RESCM, FSCM, SPARSE_SC)
Time-aware / factor models	2	2	Complete (FMA, TASC)
Staggered adoption	4	5	SDID, SpSyDiD, PPSCM, SSC ✓; SEQ_SDID queued
Spillover-aware (donor screening)	1	1	Complete (SPOTSYNTH; SpSyDiD counted under staggered)
Missing data	3	3	Complete (MCNNM, SNN, RMSI)
Identification under endogeneity	2	2	Complete (SIV, PROXIMAL)
Experimental design	5	5	Complete (LEXSCM, MAREX, SYNDES, PANGEO, SPCD)

Of mlsynth’s 36 estimators, 32 (89%) carry a strong or solid replication against their source paper or against an authoritative reference implementation. ISCM, NSC, and SPARSE_SC carry partial replications – one-draw illustrations or empirical applications without an explicit number-match – and SEQ_SDID is the single estimator still in queue.

Contributing a replication

If you have used an mlsynth estimator and replicated a paper’s number in the course of your work, please open an issue or pull request on GitHub. The two requirements are:

1. The replication target is a published number – a row of a table, a point on a figure, or a key statistic in the prose.

2. The replication is runnable from a copy of the documentation snippet plus the source dataset (CSV, R-package data, or a simulator we ship).

A reference Path A entry is Two-Step Synthetic Control’s Brooklyn-showroom section; a reference Path B entry is Time-Aware Synthetic Control (TASC)’s Section 5 grid; a reference cross-validation entry is Partially Pooled SCM (PPSCM)’s augsynth-vignette comparison.