LEXSCM — Synthetic Experimental Design (Abadie-Zhao 2026; Vives-i-Bastida 2022)

Estimator:

Lexicographic Synthetic Control (LEXSCM) — mlsynth.LEXSCM

Source:

Abadie, A., & Zhao, J. (2026), “Synthetic Controls for Experimental Design.” LEXSCM is the lexicographic solve of their program (\(\xi \to 0\): fit the treated units first, then their controls), due to Vives-i-Bastida, J. (2022), “Synthetic Experimental Design for a UBI Pilot Study.”

Replication type:

Path A — the paper’s Walmart empirical illustration — and Path B — the paper’s linear-factor simulation study.

Status:

Verified — placebo empirical and design simulation both reproduced.

LEXSCM is a design estimator (it returns a DesignResult): it chooses which units to treat before any intervention so the treated set is representative of the population and admits a valid synthetic control.

Path A — Walmart placebo design

We reproduce the paper’s empirical illustration (Sec. 4) on the Walmart store-sales panel (basedata/walmart_weekly_sales.csv — 45 stores over 143 weeks). Following the paper we design a placebo experiment with a fictitious intervention at week 129 (\(T_0 = 128\), the first ~100 weeks fitting, the rest blank, 15 experimental weeks) and m = 2 treated stores.

import pandas as pd
from mlsynth import LEXSCM

df = pd.read_csv("basedata/walmart_weekly_sales.csv")
df["candidate"] = 1
df["post"] = (df["week"] >= 129).astype(int)

res = LEXSCM({"df": df, "outcome": "sales", "unitid": "store", "time": "week",
              "candidate_col": "candidate", "m": 2, "post_col": "post",
              "frac_E": 100 / 128, "top_K": 5, "n_sims": 200,
              "n_post_grid": [5, 10, 15], "mde_horizon": "late"}).fit()
res.selected_units            # [1, 25]
res.report.att                # ~0.9% of mean sales (a placebo: near zero)

Because the intervention is a placebo, a correct design must track closely pre-period and produce an effect indistinguishable from zero:

Walmart placebo design (m = 2)

Quantity	LEXSCM	Abadie-Zhao (Sec. 4)
Pre-fit RMSE / mean sales	2.7%	small (close tracking)
Placebo effect / mean sales	0.9%	near zero
Permutation p-value	0.63	fails to reject (~0.93)
CI covers zero	yes	yes

The durable check is benchmarks/cases/lexscm_walmart.py:

python benchmarks/run_benchmarks.py --case lexscm_walmart

(LEXSCM’s lexicographic design selects stores {1, 25} and uses a moving-block conformal band, so its permutation p differs from MAREX’s MIQP design on the same panel; both deliver the same “no spurious effect” verdict.)

Path B — the design recovers the effect (Abadie-Zhao Sec. 5)

On the paper’s exact Section-5 linear-factor DGP (mlsynth.utils.marex_helpers.simulation.generate_marex_sample(), eqs 12a/12b) — J = 15 units, R = 7 observed and F = 11 unobserved covariates, T = 30 with T0 = 25 pre-intervention (T_E = 20 fitting, 5 blank, 5 experimental), sigma^2 = 1, uniform weights, every unit a candidate — we run the full experimental-design loop: LEXSCM picks the treated units from the pre-period untreated outcomes, the experiment realizes the treated potential outcome on exactly those units, and the design’s estimator is compared to the true effect. The design recovers the average effect with MAE far below its own scale, and the error shrinks moving from the single-treated-unit design to m = 2 — the paper’s Table 2 finding (“performance improves substantially when allowing m = 2 or 3”):

Design MAE relative to the effect scale (paired draws)

Treated cardinality	MAE / effect scale
m = 1 (single treated)	0.24
m = 2	0.16	(error decreases with m)

The durable check is benchmarks/cases/lexscm_design_mc.py:

python benchmarks/run_benchmarks.py --case lexscm_design_mc

It asserts the design recovers the effect (MAE well below scale at both cardinalities) and that the error decreases from m = 1 to m = 2.

References

Abadie, A., & Zhao, J. (2026). “Synthetic Controls for Experimental Design.”

Vives-i-Bastida, J. (2022). “Synthetic Experimental Design for a UBI Pilot Study.”