Ridge ASCM — Augmented Synthetic Control (Ben-Michael, Feller & Rothstein 2021)

Estimator:

Vanilla Synthetic Control (VanillaSC) — the ridge-augmentation layer (mlsynth.utils.bilevel.ridge_augment.ridge_augment_weights()).

Source:

Ben-Michael, Feller & Rothstein (2021), “The Augmented Synthetic Control Method,” JASA 116(536); reference implementation: the augsynth R package (ebenmichael/augsynth).

Replication type:

Cross-validation — mlsynth matched value-for-value to augsynth on its canonical Kansas study — and Path B — the paper’s Section-7 coverage / bias-reduction simulation.

Status:

Fully verified — empirical ladder and simulation reproduced.

Validation strategy

The Augmented SCM is a bias-correction layer, so it is validated against the authors’ own R package, augsynth, on its flagship empirical example: the effect of Kansas’s 2012 tax cuts on quarterly log GDP per capita. augsynth walks up a ladder of estimators of increasing de-biasing — plain SCM, ridge ASCM, ridge ASCM with auxiliary covariates (balanced directly), and the residualized covariate variant — and the measured effect grows while the pre-treatment imbalance falls. We reproduce that ladder cell by cell, then reproduce the paper’s Section-7 Monte Carlo (Path B).

Cross-validation — the Kansas ladder

The treated unit is Kansas (FIPS 20); treatment begins in 2012 Q2, leaving \(T_0 = 89\) pre-period quarters and \(J = 49\) donor states. The covariate model is augsynth’s documented spec,

covsyn <- augsynth(lngdpcapita ~ treated | lngdpcapita + log(revstatecapita) +
                     log(revlocalcapita) + log(avgwklywagecapita) +
                     estabscapita + emplvlcapita,
                   fips, year_qtr, kansas, progfunc = "ridge", scm = TRUE)

with each covariate transformed per row and aggregated to a pre-period mean per unit (rows carrying a missing — sparsely reported — revenue value are dropped before averaging, R’s model.frame na.omit default).

The whole ladder is reproduced through mlsynth’s public API – Augmented SCM is a mode of VanillaSC (augment="ridge"). Covariates are passed by column name; the user applies augsynth’s per-row log transforms to the DataFrame first (mlsynth’s covariate convention), and the shared-window aggregation drops a pre-period whenever any covariate is missing there, matching augsynth’s na.omit. residualize=True selects the residualized variant:

import numpy as np, pandas as pd
from mlsynth import VanillaSC

df = pd.read_csv("basedata/kansas_ascm.csv")          # long fips x quarter panel
for c in ("revstatecapita", "revlocalcapita", "avgwklywagecapita"):
    df[c] = np.log(df[c])                             # augsynth's log transforms
covs = ["lngdpcapita", "revstatecapita", "revlocalcapita",
        "avgwklywagecapita", "estabscapita", "emplvlcapita"]
base = dict(df=df, outcome="lngdpcapita", treat="treated",
            unitid="fips", time="year_qtr")

att = lambda cfg: VanillaSC({**base, **cfg}).fit().effects.att
att({})                                               # classic SCM    -0.029
att({"augment": "ridge"})                             # ridge ASCM     -0.040
att({"augment": "ridge", "covariates": covs})         # covariate ASCM -0.063
att({"augment": "ridge", "covariates": covs,
     "residualize": True})                            # residualized   -0.057

The reproduced ladder (mlsynth vs augsynth):

Specification	ATT (mlsynth)	Pre-fit L2	augsynth (ATT / L2)
Classic SCM	-0.0294	0.083	-0.029 / 0.083
Ridge ASCM	-0.0401	0.062	-0.040 / 0.062
Covariate ASCM	-0.0629	0.055	-0.061 / 0.054
Residualized	-0.0572	0.067	-0.055 / 0.067

The two no-covariate cells are exact; the covariate cells match augsynth’s values and reproduce the monotone ladder (the un-augmented SCM is the conservative end). The joint-null conformal \(p\)-value for ridge ASCM (\(0.071\)) is also reproduced to Monte-Carlo precision.

A note on the residualized penalty. After residualizing out \(K\) covariates the residual Gram is rank-deficient (\(T_0\) rows, rank \(\le J - K\)), so a cross-validation on the residuals is ill-posed and drifts to the grid floor. mlsynth tunes the penalty on the outcome scale instead — where augsynth’s residual CV lands anyway — which reproduces the published \(-0.055\) / \(0.067\) robustly.

Path B — coverage and bias reduction (Section 7)

Four data-generating processes are calibrated to the Kansas panel — a 3-factor interactive-fixed-effects model (calibrated exactly as gsynth/fect’s interFE does it with no covariates: two-way demean, then a rank-3 SVD of the residual), the same model at \(4\times\) noise, additive two-way fixed effects, and a fitted AR(3). Treatment is assigned to an extreme unit, so plain SCM struggles and the augmentation matters. Across all four DGPs ridge ASCM reduces \(|\text{bias}|\) versus plain SCM and gives near-nominal coverage (\(\approx 0.90\)–\(0.96\)), with the gain limited under high noise — the paper’s thesis.

Durable cases & tests

• ascm_kansas — the four-spec Kansas ladder cross-validated against augsynth (benchmarks/cases/ascm_kansas.py).

• augsynth_calibrated — the Section-7 coverage / bias-reduction simulation (benchmarks/cases/augsynth_calibrated.py).

• Regression tests: mlsynth/tests/test_bilevel_ridge.py (test_augsynth_kansas_replication, test_augsynth_kansas_conformal_pvalue, test_augsynth_kansas_covariate_ladder).