SBC — Synthetic Business Cycle (Shi, Xi & Xie 2025)

Estimator:

Synthetic Business Cycle (SBC) — mlsynth.SBC

Source:

Shi, Zhentao, Yishen Xi & Jin Xie (2025), “A Synthetic Business Cycle Approach to Counterfactual Analysis with Nonstationary Macroeconomic Data,” arXiv:2505.22388.

Replication type:

Path A — the authors’ German-reunification illustration — and cross-validation against their own released R code.

Status:

Fully verified, step by step, against the authors’ code.

What SBC does

Classical synthetic control matches a treated unit to a weighted average of donors on the raw outcome path. When that outcome is nonstationary — a GDP-per-capita series with a strong trend — matching on the level can lock onto a spurious comovement of trends rather than a genuine common structure. The synthetic business cycle estimator first splits every series into a slow trend and a stationary cycle with a Hamilton filter, forecasts the treated unit’s post-treatment trend from its own history, and builds a synthetic cycle from the donors. The counterfactual is the treated unit’s own projected trend plus a donor-matched cycle, so trend donors and cycle donors are allowed to differ.

This page records how mlsynth’s SBC is validated on the authors’ headline illustration — the 1990 German reunification — not against a printed table, but against the authors’ own R script run live, one step at a time.

The reference

The authors release their code at github.com/jinxi-atlas/Synthetic-business-cycle-code. Its SBC_Germany/Germany.R performs the whole procedure: a linear-projection (Hamilton) detrending via the helper lsq, a trend extrapolation via trend_predict, and the synthetic-control weight solve via Synth::synth. mlsynth’s reference bundle (benchmarks/reference/sbc_germany/) reproduces that script’s computation on the authoritative basedata/repgermany.dta (identical to the Abadie panel), and the captured outputs of each function become golden values that the unit tests in mlsynth/tests/test_sbc_reference.py pin mlsynth against.

Step-by-step agreement

The Hamilton detrending and the trend forecast match the authors’ functions to machine precision. mlsynth’s fit_hamilton_filter() reproduces lsq’s AR coefficients and cyclical residuals, for both the treated unit (detrended on the pre-treatment window) and the donors (detrended on the full sample, since the donors are untreated), to about \(10^{-8}\); forecast_treated_trend() reproduces trend_predict to the same precision. These steps are, for practical purposes, the same computation written twice.

Where the two diverge — and why mlsynth is the accurate one

The only place the two implementations disagree is the synthetic-control weight solve, and the live replication shows that the divergence is a defect in the reference solver, not in mlsynth.

At the cycle-matching step both implementations minimise the same objective over the simplex,

\[\widehat{w} \;=\; \arg\min_{w \ge 0,\; \mathbf{1}^\top w = 1} \;\bigl\lVert\, c_{1} - C\, w \,\bigr\rVert_2^2 ,\]

where \(c_1\) is the treated cycle and \(C\) the donor cycles over the effective pre-treatment window. On the German panel this program is strictly convex and well conditioned (the donor cycle matrix has full column rank, and the Gram matrix’s condition number is about \(3.8\times10^{3}\)), so its optimum is unique. Four independent solvers agree on that optimum to solver precision — mlsynth’s in-house projected-gradient routine and cvxpy’s ECOS, OSQP and SCS all attain a cyclical sum of squares of about \(1.266\times10^{6}\).

The authors’ Synth::synth (the kernlab ipop interior-point solver) instead converges to a point about \(2.6\%\) worse, a sum of squares of about \(1.299\times10^{6}\), and tightening its tolerances does not close the gap — it simply lands on a suboptimal vertex. The consequence is a visibly different weight split: ipop puts most of its mass on the Netherlands and implies an average effect near \(-1006\), while the verified optimum (which mlsynth attains) is Greece-dominant with an effect near \(-952\). The two solutions select the same donor set; they differ only because one solver reaches the optimum and the other does not.

A note on the donor labels

The authors’ shipped wide CSV permutes its donor column labels: its Japan and Portugal columns actually hold the Netherlands’ and Greece’s series (verified against the canonical repgermany.dta). So the paper’s prose naming the cycle donors as “Italy, Japan, Portugal” refers, by the correct labels, to Italy, the Netherlands and Greece — which is exactly the donor set mlsynth recovers on the correctly labelled panel. Running the reference, rather than trusting the printed names, is what surfaces this.

Verification

The durable check lives in benchmarks/cases/sbc_germany.py (the cycle weights and the 1991–1994 effect), and the per-step cross-validation in mlsynth/tests/test_sbc_reference.py (eight tests pinning each stage to the authors’ captured output):

python benchmarks/run_benchmarks.py --case sbc_germany
python -m pytest mlsynth/tests/test_sbc_reference.py

The captured reference bundle, the golden fixture, and the provenance (R and package versions, data checksums) are under benchmarks/reference/sbc_germany/; its NOTICE records the full finding. A separate Path-B Monte Carlo (sbc_mc) reproduces the paper’s simulation evidence that SBC stays competitive under cointegration.