.. _replication-sbc: SBC — Synthetic Business Cycle (Shi, Xi & Xie 2025) =================================================== :Estimator: :doc:`../sbc` — :class:`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 :class:`~mlsynth.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 :func:`~mlsynth.utils.sbc_helpers.hamilton.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 :math:`10^{-8}`; :func:`~mlsynth.utils.sbc_helpers.trend_forecast.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, .. math:: \widehat{w} \;=\; \arg\min_{w \ge 0,\; \mathbf{1}^\top w = 1} \;\bigl\lVert\, c_{1} - C\, w \,\bigr\rVert_2^2 , where :math:`c_1` is the treated cycle and :math:`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 :math:`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 :math:`1.266\times10^{6}`. The authors' ``Synth::synth`` (the kernlab ``ipop`` interior-point solver) instead converges to a point about :math:`2.6\%` worse, a sum of squares of about :math:`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 :math:`-1006`, while the verified optimum (which mlsynth attains) is Greece-dominant with an effect near :math:`-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.