Penalized Synthetic Control (Abadie & L’Hour 2021)

Estimator:

Vanilla Synthetic Control (VanillaSC) — the penalized backend (mlsynth.utils.bilevel.penalized.penalized_weights()).

Source:

Abadie & L’Hour (2021), “A Penalized Synthetic Control Estimator for Disaggregated Data,” JASA 116(536), 1817–1834; reference implementation: the authors’ pensynth repository (jeremylhour/pensynth), function wsoll1.

Replication type:

Cross-validation — mlsynth’s penalized solver matched to the authors’ own wsoll1 on identical inputs.

Status:

Fully verified — weights and ATT reproduced to solver precision.

Validation strategy

The penalized estimator adds a pairwise matching penalty to the synthetic-control objective. For treated predictors \(X_1\), donor predictors \(X_0\) and penalty \(\lambda \ge 0\) it solves (the paper’s eq. 5)

\[\min_{W}\; \lVert X_1 - X_0 W \rVert^2 \;+\; \lambda \sum_j W_j \lVert X_1 - X_{0j} \rVert^2 \quad\text{s.t.}\quad W \ge 0,\; \textstyle\sum_j W_j = 1 .\]

The penalty interpolates between the pure synthetic control (\(\lambda \to 0\)) and nearest-neighbour matching (large \(\lambda\)); by the paper’s Theorem 1, for any \(\lambda > 0\) the solution is unique and sparse. Because the program is a strictly convex quadratic program in \(W\) for \(\lambda > 0\), it has a single optimum, which makes it an ideal target for a solver-level cross-check: feed the same \((X_0, X_1, \lambda)\) to two independent solvers and they must agree.

That is exactly what the benchmark does. mlsynth’s implementation (penalized_weights(), the penalized backend of VanillaSC) solves the program by projected-gradient FISTA; the reference wsoll1 solves the identical program by the interior-point routine LowRankQP. We feed both the same predictor matrix and compare across a regularisation path.

Cross-validation — the Prop 99 path

The predictor matrix is the canonical Abadie-Diamond-Hainmueller (2010) MLAB vector: the four covariate averages — ln(personal income), retail cigarette price and percent aged 15–24 over 1980–1988, beer consumption over 1984–1988 — plus the three special lagged outcomes, cigarette sales in 1975, 1980 and 1988, matched with \(V = I\) (raw, no rescaling), the V = diag(ncol(X)) convention of the authors’ California example (examples/EXA_CaliforniaTobacco.R). It is constructed from mlsynth’s vendored basedata/augmented_cali_long.csv through mlsynth.utils.datautils.dataprep() and the covariate-mean helper VanillaSC uses — no hand-pivoting. California is the treated unit and the remaining 38 states are donors. The same \(X_0\) (\(7 \times 38\)) and \(X_1\) (\(7\)) are sent to wsoll1 and to penalized_weights over the grid \(\lambda \in \{0.001, 0.01, 0.05, 0.1, 0.25, 0.5, 1\}\).

Across the whole path the two implementations agree to solver precision: the largest donor-weight difference is \(\approx 3\times10^{-4}\) and the largest post-period ATT difference \(\approx 2\times10^{-3}\) packs. At \(\lambda = 0.1\) the synthetic California loads \(\approx 0.54\) on Idaho, with a post-1989 ATT of \(-23.4\) packs per capita matched to \(\approx 4\times10^{-4}\). With \(V = I\) (no nested \(V\) optimisation) the penalized fit is Idaho-led rather than ADH’s published Utah/Nevada pool — expected, since here the penalty, not a fitted \(V\), resolves the donor weights. As \(\lambda\) grows the weights concentrate toward the nearest neighbour (Montana), reproducing the penalty’s interpolation property.

Durable benchmark

The runnable case is pensynth_prop99 in the durable suite (benchmarks/cases/pensynth_prop99.py). It is a live cross-check: the reference wsoll1/TZero source is taken from a commit-pinned clone of jeremylhour/pensynth (benchmarks/reference/clone_pensynth.py) and LowRankQP is frozen by benchmarks/R/install_pensynth.sh, so the same solver runs every time. The case skips itself when Rscript, LowRankQP or the clone is unavailable, so it is a no-op where the reference toolchain is absent. Run it with

bash benchmarks/R/install_pensynth.sh        # one-time: R + LowRankQP
python -m benchmarks.run_benchmarks --case pensynth_prop99