SRC — Synthetic Regressing Control (Zhu 2023)

SRC — Synthetic Regressing Control (Zhu 2023)#

Estimator:

Synthetic Regressing Control (SRC) — mlsynth.SRC

Source:

Zhu, Rong J. B. (2023), “Synthetic Regressing Control,” arXiv:2306.02584 [SRC2023].

Replication type:

cross-validation against the author’s reference R implementation (Code_SMC.R), plus Path A — the Basque / ETA study.

Status:

verified — the weight computation matches the reference to machine precision.

Validation strategy#

SRC ships an R reference implementation (Code_SMC.R): a function SMCV that, given a matching matrix and predictor weights, computes the per-donor univariate coefficients \(\widehat{\theta}_j\), the plug-in \(\widehat{\sigma}^2\), and the Mallows / \(C_p\) box weights via quadprog::solve.QP. That function is the ground truth. We reproduce it two ways: a cell-by-cell numeric match of the weight computation, and the Basque counterfactual it produces.

Cross-validation — machine precision#

We build the Abadie-Gardeazabal Basque matching matrix through Synth::dataprep (the reference’s own path) and run SMCV with the predictor weights fixed to one, so the oracle is deterministic. The mlsynth weight computation mlsynth.utils.src_helpers.src_weights() is fed the identical matrix. Every quantity agrees:

Quantity

max |mlsynth − R|

\(\widehat{\theta}_j\) (all donors)

5.3e-15

box weights \(\mathbf{w}\)

1.9e-14

combined \(\widehat{\theta}_j w_j\)

2.0e-14

bias

6.3e-14

\(\widehat{\sigma}^2\)

2.4e-14

counterfactual (43 years)

1.7e-13

The synthesis QP is the load-bearing step. mlsynth solves it with an exact active-set box solver (mlsynth.utils.src_helpers.solver.solve_box_qp()), the box-[0, 1] analogue of the repository’s simplex active-set solver. On the Basque problem it reproduces solve.QP to 2e-14 with a KKT residual below 1.5e-14 — tighter than solve.QP’s own residual — and pins the box bounds exactly, so a dropped donor’s weight is exactly zero. Against a first- order QP (OSQP) it is an order of magnitude faster at synthetic-control donor sizes; against an interior-point QP (CLARABEL) it agrees on the objective but, unlike CLARABEL, leaves no donor microscopically off its bound. The solver is fuzz-tested against an independent cvxpy oracle over hundreds of random problems (mlsynth/tests/test_src.py).

Path A — the Basque study#

Run through the public estimator on basedata/basque_data.csv (outcome-only matching, treatment in 1970), SRC reproduces the Abadie-Gardeazabal result:

import pandas as pd
from mlsynth import SRC

df = pd.read_csv("basedata/basque_data.csv")
df["treat"] = ((df["regionname"] == "Basque Country (Pais Vasco)")
               & (df["year"] >= 1970)).astype(int)
res = SRC({"df": df, "outcome": "gdpcap", "treat": "treat",
           "unitid": "regionname", "time": "year",
           "display_graphs": False}).fit()

gives a pre-period RMSE of about 0.048, a mean post-1969 ATT of about \(-0.858\), and a 1997 gap of about \(-0.848\) (thousand-1986-USD per capita), with the combined donor coefficients concentrated on Murcia (\(0.63\)), Madrid (\(0.37\)) and Castilla y León (\(0.24\)). The divergence traces the familiar economic cost of ETA terrorism. The durable case is benchmarks/cases/src_basque.py.

The covariate / predictor-weight variant (Table 5)#

The paper’s Basque tables use the covariate-augmented variant (Algorithm 3) with an Abadie predictor-weight (\(V\)) search. mlsynth exposes it as a seeded opt-in — covariates with covariate_windows, the fit_window for the outcome rows, and v_search="de". Configured this way the estimator rebuilds the paper’s matching matrix from the raw panel and reproduces its Basque result:

res = SRC({
    "df": df, "outcome": "gdpcap", "treat": "treat",
    "unitid": "regionname", "time": "year",
    "covariates": [...],                 # the 12 Abadie predictors
    "covariate_windows": {...},          # 1964-69 schooling/invest, 1961-69 sectors
    "fit_window": (1960, 1969),          # time.optimize.ssr
    "v_search": "de", "v_seed": 0,
}).fit()

gives Rioja-dominant weights with Madrid second and an ATT reaching \(\approx -2.4\) by 1997 — the Table 5 / Figure 1 donor structure and magnitude, in place of the outcome-only default’s Murcia / Madrid / Castilla y León and \(-0.86\).

Why it is opt-in, and what not to over-read. The \(V\) optimum is not identified: on the Basque matching matrix a large manifold of \(V\) achieves essentially the same pre-outcome fit while producing post-period effects that range over a wide band, so the exact split among the top donors is not a well-defined function of the data — a global differential-evolution search reproduces the paper’s average placebo MSPE (Table 4) but not its per-region cells, and the split shifts with the seed. The search is therefore seeded (a given call is reproducible) and opt-in, leaving the \(C_p\)-identified Algorithm 1 as the default. Read the v_search weights as one draw from the paper’s non-identified \(V\) manifold, not as identified quantities.