jackknife+ for ridge ASCM – against augsynth on the Kansas panel

Vanilla Synthetic Control (VanillaSC)’s inference="jackknife_plus" is cross-validated against the authors’ own R package, ebenmichael/augsynth, pinned at commit 7a90ea48 (augsynth 0.2.0). The reference is not vendored; install it with bash benchmarks/R/install_augsynth.sh and regenerate the comparison with Rscript benchmarks/reference/ascm_jackknife_plus/reference.R. The captured gold is committed, so the benchmark and the tests run without an R toolchain.

The setup is augsynth’s own Kansas tax-cut example: quarterly log GDP per capita, 50 states, treatment at 2012 Q2, 89 pre-treatment periods and 16 post, ridge-augmented synthetic control with no auxiliary covariates.

What the procedure is

Despite the name, this is not the jackknife-plus of Barber, Candès, Ramdas and Tibshirani, which resamples over observations, and it is not the delete-one-unit jackknife that PPSCM — augsynth multisynth (Paglayan collective bargaining) uses. It resamples over pre-treatment periods.

Each pre-period is held out in turn, the augmented control is refitted on the remaining pre-periods, and the refit is asked to predict the period it never saw. A period the fit cannot recover when hidden is evidence that the post-treatment prediction is also uncertain, so the interval is built from those held-out errors. With 89 pre-periods that is 89 refits for one interval.

What matches, seam by seam

Because Ridge ASCM — Augmented Synthetic Control (Ben-Michael, Feller & Rothstein 2021) already pins the ridge ASCM point estimate against a live augsynth run on this same panel, the fit is not a confound here and everything below isolates the inference.

quantity	agreement
held-out error, per dropped pre-period	89/89, worst 3.5e-8
refit counterfactual path, per dropped pre-period	89 x 17, worst 1.7e-7
assembled bounds, default branch	worst 3.8e-9
assembled bounds, conservative branch	worst 1.8e-7
bound on the post-period average	6.4e-10 (lower), 3.2e-9 (upper)

The per-drop rows are pinned as well as the assembled ones, deliberately. The interval alone would report that a regression happened; the per-drop seam reports which refit caused it.

Why the two branches earn different tolerances

The base simplex fit underneath every refit is a constrained quadratic program, solved here by mlsynth’s own exact solver and in the reference by quadprog and osqp. Two exact solvers agree to their own tolerances rather than bit-for-bit, so roughly 1e-7 is the floor for any quantity downstream of the QP.

The default branch takes quantiles across all 89 drops, which averages that difference away and reaches 4e-9. The conservative branch instead takes the minimum and maximum over drops, which deliberately selects the single most extreme refit – exactly where the solver difference is largest – and lands at 1.8e-7. Looser is correct there rather than lax: an interval built from order statistics cannot be more reproducible than its worst-case term.

Two findings worth keeping

The penalty is pinned across refits, and this is not obvious from the paper. augsynth.R stores the fitted ridge penalty into extra_args before any resampling (extra_args$lambda <- augsynth$lambda), so every held-out refit reuses the original fit’s lambda and none re-runs cross-validation. Re-selecting per drop would give 89 different penalties and a materially different interval, and would multiply the cost by a CV sweep per refit. Read from the source, not inferred from the output.

conservative = TRUE is not reliably conservative. It widens by the \(1-\alpha\) quantile of the absolute held-out errors, where the default branch reaches the \(1-\alpha/2\) quantile of \(\widehat{y}^{N}_{1t}(j) + |e_j|\). When the held-out errors are right-skewed – on this panel the largest absolute error is 0.037 against a 95th percentile of 0.013 – the default’s deeper reach into that tail dominates the wider min/max envelope, and the “conservative” interval comes out tighter. It is narrower at 16 of the 17 reported periods here. mlsynth reproduces this rather than imposing an ordering the reference does not have, and the benchmark pins the branch’s construction instead of a width comparison so that nobody later “corrects” it.

Verification

Pinned in benchmarks/cases/ascm_jackknife_plus.py (10 rows) and in mlsynth/tests/test_ascm_jackknife_plus.py (31 tests), which asserts the per-drop seam before the assembled bounds so a failure localises.