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-five of the thirty-six estimators are currently fully verified; ISCM carries a one-draw illustration only (its paper relies on a non-public panel and provides no Monte Carlo to reproduce).

The runnable cases that pin these numbers – and guard against regressions – live in the durable benchmark suite; see Benchmarks for the catalogue and how to run them.

• 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. Replications are progressively being broken out into dedicated pages (one per source paper) collected in the toctree below; the catalogue entries link to a dedicated page where one exists.

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. → dedicated page: TSSC — Two-Step Synthetic Control (Li & Shankar 2024); durable cases tssc_brooklyn (Path A) and tssc_figure2 (Path B).

• Vanilla Synthetic Control (VanillaSC) – Standard SC (ADH 2010/2015; Abadie-Gardeazabal 2003). Path A: the three canonical studies on their original datasets – Prop 99 (Utah/Nevada/Montana/Colorado/Connecticut, ATT \(\approx -19\) packs), German reunification (Austria-dominant donor pool), Basque (Cataluna \(\approx 0.8\) + Madrid \(\approx 0.2\), ATT \(\approx -0.68\)); plus the Lei-Sudijono (2025) leave-two-out Table-1 p-value relations. The Prop 99 fit reproduces ADH 2010 Table 2 with the full predictor spec (Utah 0.335, Nevada 0.236, Montana 0.202, Colorado 0.160, Connecticut 0.068, each within 0.005 of the paper; durable: vanillasc_prop99). → dedicated page: VanillaSC — Standard Synthetic Control (ADH 2010/2015; Abadie-Gardeazabal 2003).

• ascm_kansas – Ben-Michael, Feller & Rothstein (2021) Augmented SCM (the ridge-augmentation layer on VanillaSC). Cross-validation vs augsynth: the canonical Kansas tax-cut ladder reproduced value-for-value – classic SCM (ATT \(-0.029\)), ridge ASCM (\(-0.040\)), covariate ASCM (\(-0.061\)) and residualized (\(-0.055\)), pre-fit \(L_2\) falling \(0.083 \to 0.054\); the ridge conformal \(p\)-value (\(0.071\)) also matches. Path B: the Section-7 thesis – near-nominal coverage (\(\approx 0.90\)–\(0.96\)) and bias reduction across four Kansas-calibrated DGPs. Status: done. → dedicated page: Ridge ASCM — Augmented Synthetic Control (Ben-Michael, Feller & Rothstein 2021); durable cases ascm_kansas and augsynth_calibrated.

• Matching and Synthetic Control (MASC) – Kellogg, Mogstad, Pouliot & Torgovitsky (2020) matching + synthetic control. Path A: the KMPT Section-5 Basque Country / ETA-terrorism study on basque_jasa.csv, reproduced value for value when run as the authors did (MSCMT/synth SC + covariate matching): the CV selects pure SC (\(\hat\varphi = 0\), as in KMPT), Cataluna + Madrid carry all the weight (1.00 vs 0.85 + 0.15), pre-RMSE \(\$89\) vs \(\$94\), and ATT \(-\$585\) vs the paper’s \(-\$580\). The sc_backend and match_on toggles expose both optimiser/matching choices (durable: masc_basque).

• Forward Difference-in-Differences (FDID) – Li (2024) Forward Difference-in-Differences. Path A: the author’s public Hong Kong GDP companion replication reproduced cell by cell (FDID ATT \(0.0254\), \(53.84\%\), pre-\(R^2 = 0.843\), 9 of 24 controls). 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 \(0.082\). → dedicated page: FDID — Forward Difference-in-Differences (Li 2024).

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 (with ridge="sdid" + a refined rho grid; durable: hsc_hongkong). Path B: Liu-Xu regime-adaptation grid with ARIMA(1,1,0) trends – HSC tracks the oracle-best fixed method in both regimes (rho adapts; durable: hsc_mc).

• Synthetic Business Cycle (SBC) – Synthetic Business Cycle. Path A: German reunification – Greece weight 0.44, Netherlands 0.37, \(\widehat{\mathrm{ATT}} \approx -952\) (durable: sbc_germany); also reproduces the Hong Kong handover. Path B: Shi-Xi-Xie Models 1-3 nonstationary DGP, post-MSE ratio < 1 in every model and highest (SC competitive) under cointegration (durable: sbc_mc).

Generalising the estimand, treatment, or unit

• Synthetic Control with Multiple Outcomes (SCMO) – multi-outcome SC, both variants. Path A: Tian-Lee-Panchenko (2026) German reunification (nine pre-1989 indicators) – the concatenated synthetic reproduces their Table 2 balance cell by cell (synthetic 1989 GDP per capita \(19029.8\); CPI \(3.1\); trade \(59.1\); tax \(34.1\)), pre-RMSE \(= 110\) (durable: scmo_germany). Path B (concatenated): TLP Table 1 == Sun et al. Simulation1.R – bias falls and pre-fit rises with the outcome count \(K\) across \(T_0 \in \{1, 5, 10\}\) (durable: scmo_concatenated_mc). Path B (averaged): Sun-Ben-Michael-Feller (2025) Appendix-D regime contrast – averaging beats the separate SC under a common factor and hurts under purely idiosyncratic factors (durable: scmo_averaged_mc). → dedicated page: SCMO — Synthetic Control with Multiple Outcomes (Tian et al. 2026; Sun et al. 2025).

• Relaxed / Penalized Synthetic Control (RESCM) – SCM-relaxation (Liao, Shi & Zheng 2026). Path A: Brexit / UK real GDP – the L2 relaxation’s cumulative GDP loss is \(\approx 4.0\%\) (paper 3.85%, treatment 2016Q3), dense weights on the major EU economies that classic SC drops (durable: rescm_brexit). Cross-validation: mlsynth’s L2 relaxation matches the authors’ scmrelax package cell by cell at a matched \(\tau\) (donor-weight \(L_1\) distance \(\approx 0.0014\); durable: rescm_relax_ref). Path B: the Section-5 latent-group Monte Carlo – with \(J \gg T_0\), the L2 relaxation’s out-of-sample error against the oracle counterfactual is \(\approx 0.43\) of classic SC’s (median; paper’s Table 1 \(\approx 0.15\)–\(0.53\)), beating SC in \(\approx 73\%\) of reps (durable: rescm_relax_mc). → dedicated page: RESCM — SCM-relaxation (Liao, Shi & Zheng 2026).

• Relaxed / Penalized Synthetic Control (RESCM) – L-infinity-norm SC (Wang, Xing & Ye 2025; LINF / L1LINF). Cross-validation: mlsynth’s L-infinity engine matches the authors’ LinfinitySC code cell by cell in the unique \(T_0>J\) regime (donor-weight \(L_1\) distance \(\approx 0.0019\); durable: linf_crossval_ref). Path A: Proposition 99 – dense weighting across all 38 donors vs classic SC’s 6 (durable: linf_prop99). Path B: the Section 5 two-factor Monte Carlo – L-infinity beats sparse SC in the dense DGPs, L1LINF in the sparse one (durable: linf_sim). → dedicated page: LINF / L1LINF — L-infinity-norm SC (Wang, Xing & Ye 2025).

• 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 \(\approx 0.80\) (durable: ctsc_powell_mc).

• Synthetic Historical Control (SHC) – Chen, Yang & Yang (2024) Synthetic Historical Control. Path B: the Section-3.1 effect-free simulation – with \(\delta_t = 0\), SHC recovers the latent confounder \(\ell_t\) to \(\approx 10^{-3}\) matching error pre and post (MSE_pre / MSE_post near zero, rising only mildly with the horizon, as the bias bound predicts; durable: shc_recovery_mc).

• 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. Cross-validation: matched to Tian’s (2023) own R implementation on the Proposition 99 panel (Table 2) – per-donor NSC weights track the author’s to correlation 0.989 (max \(|\Delta| = 0.024\)), recovering the same signed donor pool, with an average effect \(-19.1\) and an effect path (\(-9.1/-22.6/-27.0\) in 1990/1995/2000) matching the paper’s \(-9.5/-24.5/-28.7\). Path B: the nonlinear-outcome Monte Carlo (Section 4, Table 1) – on the \(r = 2\) panel NSC’s 95% CI covers near nominal (\(0.93/0.97\) at \(J = 25/50\), paper \(0.935/0.950\)) and the error shrinks as the donor pool grows. → dedicated page: NSC — Nonlinear Synthetic Control (Tian 2023); durable cases nsc_prop99, nsc_mc.

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 0.485, France 0.354, pre-RMSE \(\approx 89\) matching Bayani’s reference figures (durable: clustersc_rpca_germany). Path B: Amjad-Shah-Shen periodicity DGP plus a two-factor DGP for missing-data robustness. Path B (subgroups, cross-validated): Rho et al. (2025) ClusterSC – in the high-dimensional-subgroup regime (pooled rank \(>T_0\)) ClusterSC beats whole-pool RSC at every noise level (MSE down \(60.8\% / 43.2\% / 24.3\%\) at \(\sigma = 0.10/0.25/0.40\)), and the authors’ own code reproduces its \(\approx 50\%\) headline on its own DGP. Path B (RSC, Amjad-Shah-Shen 2018): the PCR-SC training error approximates its generalization error (gen/train \(\approx 1.0\text{-}1.15\)) across noise, the RSC Table-1 finding. Cross-validation (Shen et al. CIs): mlsynth’s variance code matches deshen24/panel-data-regressions to machine precision, and the doubly-robust CI attains \(\approx 95\%\) coverage for all three estimands where a single-source CI drops to \(63\%\). → dedicated page: ClusterSC — Synthetic Control with Donor Selection (Rho et al. 2025).

• Multi-Level Synthetic Control (mlSC) – Bottmer (2024) multi-level SC. Cross-validation against the author’s mlSC_estimator (multi-level-sc-estimator, pinned commit 0fb2639) on the README hierarchical-factor DGP. Path A (one draw): \(\widehat{\mathrm{ATT}} = +0.011930\) vs the reference \(+0.011928\), \(\lambda = 1.970185\) to machine precision. Path B (200 independent draws): both unbiased for the true zero ATT and in near-machine agreement, \(\max |\Delta| = 5.76 \times 10^{-4}\). The case clones the reference at run time and skips gracefully when it is unavailable (durable: mlsc_bottmer).

• Panel Data Approach (PDA) – Shi & Huang (2023) panel data approach. Path A: Hong Kong economic integration (HCW 24-economy panel, quarterly GDP growth) – all three methods recover a positive, highly significant CEPA effect, with the L2-relaxation estimate (2.48%, \(t \approx 7.7\)) reproducing Shi & Wang’s Appendix-E.1 headline of \(+2.65\%\) (\(t = 8.35\)); LASSO and forward selection bracket it at 3.3% and 3.9% (durable: pda_hongkong). Path B: Table-1 size/power geometry on the four-factor DGP – forward selection stays parsimonious (~4 donors) and correctly sized (≈ 0.08-0.09) under both factor structures, while LASSO over-selects (9-15) and its size inflates under dynamic factors (0.14 vs 0.065 i.i.d.), matching the paper; both fully powered at D5 (durable: pda_table1). mlsynth’s LASSO is cross-validated, not the paper’s modified-BIC, so its LASSO cells are a CV variant, not a cell-by-cell match. Path B (LASSO): Li & Bell (2017) Table 2 on a dense three-factor DGP with \(N=31 > T_1=25\) – the LASSO control-unit selection stays parsimonious (~7 of 30) and its out-of-sample PMSE scales with the idiosyncratic noise (\(\approx 1.5 / 0.97 / 0.18\) at \(\sigma^2 = 1 / 0.5 / 0.1\), vs the paper’s \(1.77 / 0.96 / 0.22\); durable: pda_lasso_sim). Path B (l2): Shi & Wang (2024) Table 2 on a dense strong-factor DGP (\(N=100\)) – the L2-relaxation test’s size shrinks toward nominal as \(T_1\) grows (\(0.20\) at \(T_1=50\) → \(0.10\) at \(200\), vs the paper’s \(0.142 \to 0.072\)) and its power matches (\(0.567\) vs \(0.570\) at \(T_1=50\); durable: pda_l2_sim). Path A (fsPDA): the China luxury-watch study (anti-corruption campaign, Jan-2013) on china_watches_long.csv – forward selection picks 3 controls, in-sample \(R^2 = 0.777\) and monthly ATE \(-3.09\%\), matching Shi & Huang’s Section 5 cell-for-cell; with the prewhitened Newey-West variance from their application script the effect is significant (\(t = -2.51\) vs the paper’s \(-2.457\); durable: pda_luxurywatch). Path A (l2): the Shi & Wang (2024) China PPI / real-estate-regulation study on china_ppi_long.csv – with the authors’ standardisation the L2-relaxation recovers a large significant negative monthly PPI effect, \(-5.95\%\) (\(t = -4.48\)), against their \(-6.40\%\) (\(t = -3.61\)); the gap is mlsynth’s time-respecting sequential CV vs the paper’s future-leaking 5-block K-fold (durable: pda_ppi). Path A (multi-treat): the Shi & Wang (2024) Brexit study on brexit_long.parquet – 52 UK firms treated as a cross-section against 300 controls; on the first post-referendum day the L2-relaxation per-period ATE on UK returns is \(-4.50\%\) (\(t = -7.5\)), reproducing the paper’s \(-4.31\%\) (\(t = -7.39\)). All 52 per-firm fits run through one shared OSQP factorisation (durable: pda_brexit).

• 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 (durable: fscm_prop99).

• Sparse Synthetic Control (SparseSC) – Sparse SC. Path A: California Proposition 99 with an over-rich augmented predictor set, with conformal inference – the L1 penalty prunes to 6 of 33 predictors and the effect lands at \(-17.9\) packs (95% CI \([-21.3, -15.4]\)) on the ADH donor pool (durable: sparse_sc_prop99). See the dedicated page.

• Multivariate Square-root Lasso Synthetic Control (MSQRT) – Shen, Song & Abadie (2025) multivariate square-root-Lasso SC for high-dimensional disaggregated data. Path B: the Section-6 simulation (true ATT = 0) – the estimator is unbiased (mean bias within Monte-Carlo noise of zero) and the imputed-counterfactual RMSE stays near the \(\sigma = 0.5\) noise floor and roughly flat in the number of treated units \(m\), the property that makes the many-treated regime work (durable: msqrt_sim; run at a reduced \(n = 80\) regime – the paper’s exact \(n = 400\) RMSE level is \(\approx 0.72\)).

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 – and stays near nominal across the equal / treated-smaller / treated-larger variance regimes where Xu’s interval fails (durable: fma_coverage_mc).

• 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: the Section-5.2 \((q, r)\) state-space ablation (Figures 3-4) – TASC has the smaller median counterfactual RMSE than vanilla SC in all four regimes (ratio 0.84-0.93), strongest under high observation noise, the paper’s headline (durable: tasc_mc).

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 \(8.4\). Cross-validation: matched to causaltensor.SDID on the same matrix to \(|\Delta| = 3.1\times 10^{-3}\). → dedicated page: SDID — Synthetic Difference-in-Differences (Arkhangelsky et al. 2021).

• Spatial Synthetic Difference-in-Differences (SpSyDiD) – Spatial Synthetic-DiD. Path B (cross-validation): the authors’ State-Level Monte Carlo (serenini/spatial_SDID) run per-rep against their own algorithm – per-rep ATT correlation 0.996, mean ATT bias \(\approx 0.02\) for both implementations (the paper’s headline unbiasedness). → dedicated page: SpSyDiD — Spatial Synthetic-DiD (Serenini & Masek 2024).

• 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 (cross-validation): the Guanajuato police-reform application (Section 4; \(N = 33\), 10 staggered adopters), cross-validated against the authors’ committed reference output (jcao0/staggered_synthetic_control, pinned commit 74e77d4) – event-time ATT estimates match 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, and the smallest SSC Gram eigenvalue per outcome matches the reference Table1_eigenvalue.csv to \(\approx 10^{-4}\) (durable: ssc_guanajuato). 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.

• Spillover-Aware Synthetic Control (SPILLSYNTH) – spillover-aware SC, a four-method dispatcher, each method benchmarked. Cao-Dowd (method='cd'), Path A cross-validation: Proposition 99 with spillover on the 51-unit panel (50 states + DC, 1970-2000), cross-validated against the authors’ committed MATLAB output (jcao0/synthetic-control-spillover, pinned commit 60bbebe) – the California spillover-adjusted ATT path reproduces spillover.csv to \(\approx 10^{-4}\); the spillover-adjusted ATT (avg \(-9.44\)) is attenuated relative to vanilla SCM (\(-10.81\)) and near zero in the first four post years (\(-0.85\)) where spillover to Nevada/Oregon/DC inflates the classical estimate (durable: spillsynth_prop99). Inclusive SCM (method='iscm', Di Stefano & Mellace 2024), Path A: German reunification keeping Austria in the donor pool, the outcome-only inclusive fit (the no-covariates regime Melnychuk’s reference also reports; distinct from the Iterative SCM and from the with-covariates Table-2 spec) – Austria carries \(\approx 0.33\) of synthetic West Germany and West Germany \(\approx 0.32\) of synthetic Austria (\(\det\Omega \approx 0.90\)), and the inclusive ATT is more negative than the naive gap (durable: spillsynth_iscm_germany). Grossi et al. (method='grossi', 2025), Path A: the direct/spillover decomposition on German reunification – a direct effect (\(\approx -1605\)) more negative than the naive gap, with Austria removed from the donor pool but assigned a negative spillover (durable: spillsynth_grossi_germany). SAR Bayesian (method='sar', Sakaguchi & Tagawa 2026), Path B (cross-validation): mlsynth runs the authors’ own spatial-AR simulation DGP (rook lattice, sigma2=1) and reproduces their published Section-5.2 cells (N=16, T0=20) – the proposed method is unbiased at every \(\rho\) (paper \(|\text{bias}|\le 0.003\)) with ~95% coverage, while ordinary SCM’s bias grows with the spillover (mlsynth \(+0.10/+0.57\) vs the paper’s \(+0.092/+0.504\) at \(\rho=0.3/0.8\)); the proposed estimator de-biases SCM and nests it at \(\rho=0\) (durable: spillsynth_sar_mc).

• Sequential Synthetic Difference-in-Differences (Sequential SDiD) – Arkhangelsky & Samkov (2025) Sequential Synthetic DiD. Path B: the Section-5.2.2 calibrated-panel Monte Carlo (Table 1) reconstructed from the paper’s description (the authors’ CPS log-wage panel is not public) – under an interactive-fixed-effects violation of parallel trends, standard DiD’s 95% CI coverage collapses (\(\approx 0.45\), paper \(\approx 0.70\)) while Sequential SDiD stays near nominal (\(0.945\)) with roughly five-times-smaller bias and lower RMSE – the paper’s “DiD severely biased, Sequential SDiD reliable” headline (durable: seq_sdid_mc). A noiseless rank-one IFE corollary pins exact machine-precision recovery for every donor-balanced cohort. → dedicated page: SEQ_SDID — Sequential Synthetic DiD (Arkhangelsky & Samkov 2025).

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. Cross-validation: matched to causaltensor’s MC-NNM ATT (-19.83 vs -20.27, \(\approx\) 2%; estimand-level, given the differing FE sub-solvers). → dedicated page: MCNNM — Matrix Completion with Nuclear-Norm Minimization (Athey et al. 2021).

• 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, with SIV’s debiasing far below 2SLS at every level (durable: siv_syria_mc).

• Dynamic Synthetic Control for Auto-Regressive processes (DSCAR) – Zheng & Chen (2024) dynamic SC for auto-regressive processes. Path A: Beijing heavy-pollution alerts on station-level PM2.5 – the orange alert reproduces the paper value for value (ATT \(-33.78\) vs \(-33.8\), a 24.3% reduction; counterfactual 139.1 vs 139.0, treated mean 105.3 vs 105.3), while the red alert recovers a large negative effect (\(-55.7\), ~22% reduction) whose magnitude differs from the paper’s \(-70.4\) (the released code omits a per-unit de-meaning step), pinned as a qualitative + regression guard (durable: dscar_beijing).

• Proximal Inference Synthetic Control (PROXIMAL) – Proximal SC. Path A (cross-validation): Panic of 1907 on the Trust panel, cross-validated against the authors’ Python reference (freshtaste/proximal, pinned commit a67d81e) and the papers’ Table 3 – PI \(-1.148\) vs. reference \(-1.138\); PIS \(-1.148\) vs. \(-1.134\); PIPost \(-1.220\) (matching the reference to 1e-8); the no-proxy SPSC (\(-0.892\)), doubly-robust DR (\(-1.194\)) and PIPW (\(-0.854\)) are pinned alongside (durable: proximal_panic1907). Path B (PI/PIS/PIPost): the authors’ freshtaste/proximal surrogates DGP (Sec. 4.1, trending \(\log t\) factor, \(\mathrm{True.ATT}=1\)) – the proximal trio recovers the truth with near-nominal coverage and PIS attains the lowest MSE, while classical SC is biased by the trend (durable: proximal_surrogates_mc). Path B (SPSC): the authors’ own interactive-fixed-effects Monte Carlo (the qkrcks0218/SPSC README DGP, \(\mathrm{True.ATT}=3\)) – over 60 draws SPSC recovers the truth essentially without bias under either detrending choice, and the detrended SPSC-DT covers near the nominal 95% while the un-detrended SPSC-NoDT under-covers under the trend (durable: spsc_ifem_mc). Path B (DR/PIPW): the authors’ DR_Proximal_SC/simulation/normal design (\(\mathrm{True.ATE}=2\)) – DR and PIPW recover the truth with near-nominal Wald coverage, and DR is robust to an outcome-bridge misspecification that collapses the outcome-only PI (\(\approx 4.2\) vs. DR \(\approx 2.0\); durable: dr_proximal_mc).

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 \(R^2 \in [0.90, 0.98]\) (durable: pangeo_supergeo_mc).

• 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 synthetic experimental design (Abadie-Zhao 2026; Vives-i-Bastida 2022). Path A: the Walmart 45-store placebo design – pre-fit RMSE \(\approx 2.7\%\) of mean sales, placebo effect \(\approx 0.9\%\), permutation p \(\approx 0.63\) (CI covers zero), the paper’s “no spurious effect” result. Path B: the Abadie-Zhao Section-5 linear-factor design simulation (exact params: \(J=15\), \(T_E=20\)) – the design recovers the planted effect with MAE/scale \(\approx 0.24\) for the single-treated-unit design falling to \(\approx 0.16\) at \(m=2\) (Table-2 monotonicity). → dedicated page: LEXSCM — Synthetic Experimental Design (Abadie-Zhao 2026; Vives-i-Bastida 2022).

• GeoLift Market Selection (GEOLIFT) – Meta’s GeoLift market-selection design (augsynth Augmented SCM + conformal inference). Cross-validation vs GeoLift/augsynth: the GeoLift_Walkthrough realized report reproduced value-for-value with fixed_effects=True – per-unit ATT 155.6, percent lift 5.4%, summed incremental 4667, conformal \(p = 0.01\). The match needs unit fixed effects (per-unit demeaning), a mean-of-units fit target, the all-period conformal refit, and augsynth’s period-space ridge ASCM; the conformal test is shown calibrated by a 40-market placebo study. Status: done. → dedicated page: GEOLIFT — Meta’s GeoLift walkthrough (augsynth cross-validation); durable case geolift_walkthrough (plus test_geolift_walkthrough).

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	1	2	NSC ✓ (cross-validated vs author’s R + Path-B MC); ISCM one-draw only (non-public panel, no MC)
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	5	5	Complete (SDID, SpSyDiD, PPSCM, SSC, SEQ_SDID)
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, 35 (97%) carry a strong or solid replication against their source paper or against an authoritative reference implementation. Only ISCM remains a one-draw illustration – its paper relies on a non-public panel and provides no Monte Carlo to reproduce.

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.