Synthetic Control

Synthetic control

An opinionated synthetic-control workflow grounded in a read canon (references/canon.md, current as of 2026-07-28): Abadie's JEL survey, the practice manual by the method's originator, with the synthetic-DiD bridge and the DiD-vs-SC boundary supplied by the Arkhangelsky-Imbens panel survey (shared with the did skill). Deliverable: the recommendation with its citation, the R estimation and diagnostics code, and a methods paragraph.

Refresh path: run litreview on the method since the canon date, then propose additions to references/canon.md as flagged addenda.

Which precedent your design looks like

references/details.md carries what each case is the precedent for.

The feasibility gate: is SC usable here at all?

The method's originator is explicit that mechanical applications are risky and that there are situations where the honest answer is to walk away. Check before estimating:

• One or a few treated AGGREGATE units (state, market, country, category), a donor pool of genuinely comparable untreated units, and outcomes that co-move across units.
• A long pre-period. But a long T0 cannot repair a bad fit: the bias bound is derived under (near-)perfect predictor fit, and when pre-period fit is poor the recommendation is to not use synthetic control, full stop.
• Enough post-intervention periods. Abadie, Diamond, and Hainmueller (2015) also require a sizable number of them when the effect emerges gradually after the intervention or changes over time, so a long pre-period with two post-periods does not pass this gate.
• The converse trap: good pre-period fit with a short T0 or a noisy outcome can be spurious overfitting on transitory shocks, and a larger donor pool makes overfitting easier, so a bigger J is not automatically better.
• The expected effect must be large relative to unit-specific outcome volatility. Compare the effect you expect against the treated unit's pre-period residual volatility (for instance the pre-period RMSPE, or the outcome's detrended SD) and report that comparison. Sales and attention data are volatile, and when the effect is not clearly larger, pre-filter unit-specific noise (denoising a la robust SC) or concede the effect is undetectable. The judgment and its basis go in the writeup.
• No anticipation, or shift T0 earlier than any plausible anticipation (harmless if too early, because the effect path is unrestricted). Forward-buying before an announced price change is the marketing version of the problem.
• No interference: donors exposed to the treatment (neighboring markets, national campaign spillover) are handled in design (drop them, check the fit cost) or kept with the bias signed and the estimate reported as a bound.
• The treated unit should be plausibly inside the donors' convex hull; weights are nonnegative and sum to one, so SC never extrapolates. Extreme treated units in levels need an outcome transformation (differences, growth rates, pre-mean deviations) or SDID, and differencing inflates the noise share of variance, which raises overfitting risk.

When the gate fails, say so and decline to estimate; route back to causal-design. When the gate fails on pre-period length or convex-hull grounds, check the extensions map (forward DiD, augmented DiD, HCW) before declining and routing back.

Donor pool discipline

Exclude donors treated with similar interventions in the window, donors hit by large idiosyncratic shocks that would not have hit the treated unit, and donors dissimilar on observed or suspected unobserved attributes. The canon's line: including units the analyst regards as unsuitable controls "is a recipe for bias." The worked precedents: dropping US states with their own tobacco programs (Prop 99), restricting the reunification donor pool to OECD economies.

Estimator anatomy and researcher degrees of freedom

The counterfactual is a convex combination of donors chosen so the synthetic unit matches the treated unit's pre-intervention predictors, with predictor-importance weights V. The constraints buy transparency and sparsity: at most k donors get positive weight, and you can name them. Panel regression on the same data is implicitly a synthetic control whose weights sum to one but can be negative, so it extrapolates silently (four donor countries get negative regression weights in the reunification example).

The degrees of freedom, each with a discipline:

• Predictors: pre-intervention outcomes PLUS substantive covariates. Pre-outcomes alone push excluded covariates into the unobserved loadings and raise the bias bound. The Mixtape (Cunningham, Causal Inference: The Remix, synthetic-control chapter) reports that researchers increasingly rely solely on lagged outcomes as covariates (Ben-Michael, Feller, and Rothstein 2021); this skill rejects the drift because the excluded covariates are what inflate Abadie's bias bound. The exception is an outcome series known to be driven by strongly co-moving common factors, which is why one pre-period average sufficed for reunification.
• V: inverse-variance as the simple default; better, minimize pre-period MSPE or pick V by a training/validation split of the pre-period. Cross-validated V is not always unique, so show the estimate is stable across reasonable V choices.
• Weights are computed from pre-intervention data only, so the design can be locked, even preregistered, before post-treatment outcomes are seen; the canon compares this to a pre-analysis plan. Use that: fix the specification before looking at effects.

The boundary: DiD, SC, or synthetic DiD

One continuous decision path, not competing methods:

• DiD is the special case of the SC factor model with constant factor loadings. If the treated unit's pre-trend parallels a plausible comparison average, use did. Long pre-periods are key for SC identification. DiD needs one pre-period to identify the ATT and more only for credibility, so the SC pre-period gate does not transfer to a DiD routing decision.
• When the pre-period plot shows the donor average diverging from the treated unit before treatment, parallel trends has already failed and SC is the tool. The sharpest known routing result: under selection on lagged outcomes with autocorrelated errors, DiD is inconsistent even as the pre-period grows while SC is consistent (Arkhangelsky-Hirshberg, via the panel survey). If units select into treatment on recent outcomes, that is the SC regime.
• Synthetic DiD is SC with unit weights plus analogous time weights and a DiD-style adjustment: it does not require the pre-fit to be perfect, it differences the remaining gap out. The price is a stable-bias assumption on that gap. In simulations calibrated to real panels, SDID typically outperforms DiD, SC, and matrix completion. Practical default: run canonical SC when the gate passes cleanly; when level fit is the sticking point, move to SDID and say why. Reporting discipline: plot the time weights beside the trajectory figure so readers see which pre-years carry the counterfactual, and plot the unit weights so they see which donors do. The intercept means the treated and synthetic series will not overlay, so the visual pre-fit check canonical SC leans on is weaker here. No accepted substitute exists yet. The four assumptions SDID actually makes are in references/details.md.
• Staggered adoption with many treated units forecloses the standard SC estimator; that is did territory (or multisynth / the factor-model estimators, below). Staggered SDID is the exception: the appendix of Arkhangelsky et al. (2021), section 2.3 of the Stata Journal SDID article (Clarke, Pailanir, Athey, and Imbens 2024), and Porreca (2022). No CRAN package covers it. Porreca ships code at github.com/zachporreca/staggered_adoption_synthdid.

Inference

The primary mode is design-based permutation, honest about its limits:

• In-space placebos: reassign treatment to each donor, refit, pool the gaps. The test statistic is the post/pre RMSPE ratio, which corrects for placebos that fit badly pre-period; p is the treated unit's rank among J+1 ratios. Report the spaghetti plot of placebo gaps, never only p; the plot carries the magnitude.
• The smallest attainable p is 1/(J+1). A tiny donor pool cannot deliver conventional significance, and pretending otherwise is the error. One-sided versions of the statistic add real power in small pools.
• Uniform permutation is a benchmark, not a model of assignment; sensitivity to non-uniform assignment and test-inversion confidence sets exist (Firpo-Possebom).
• Model-based complements when intervals are wanted: conformal inference (Chernozhukov et al.) and prediction intervals (scpi). For SDID, placebo, jackknife, and bootstrap variance estimators ship with the estimator. The data-shape conditions for choosing among them are in references/details.md.

Diagnostics battery

1. Pre-period fit, the entry gate: table the treated unit's predictor values against the synthetic unit's and plot both trajectories over the whole pre-period. Visible gaps in either mean do not proceed (tighten the pool, change predictors or transformation, bias-correct, move to SDID, or abandon). The balance table has one value per variable per group, so it is a display, not a test.
2. Backdating (in-time placebo, the Heckman-Hotz preprogram test): move the intervention date back, re-estimate on pre-data only. Pass has two parts: no effect opens during the fake post-period, and the gap still opens at the true date with the same sign and shape. A pre-gap estimates the bias's direction and size. Caveat carried from the panel survey: backdating assumes strict exogeneity and backfires under selection on recent shocks (the AMA Marketing News routing source (Li, Luo, and Pattabhiramaiah 2024; 'AMA' hereafter) states this exercise as one of two mandatory best practices; the strict-exogeneity caveat here governs when it is informative).
3. In-space placebos with the RMSPE ratio (above).
4. Leave-one-out on positive-weight donors: refit dropping each in turn; the conclusion must not hinge on one donor. When it does, check whether that donor had its own intervention or shock.
5. Robustness across predictor sets, V choices, and tightened donor pools; drift as the pool tightens signals interpolation bias from dissimilar donors. Searching over pre-treatment lag choices raises the false-positive rate (Ferman, Pinto, and Possebom 2020), so pre-commit the predictor set or report every specification tried.
6. Outcome-transformation check when levels are hard to match (levels vs differences vs growth rates vs pre-mean deviations), remembering the noise-amplification tradeoff and that matching changes alone is not credible when the level itself drives dynamics.
7. Spillover accounting: with and without exposed donors; when kept, sign the bias and interpret the estimate as a bound.

Extensions, and when to reach for them

• Imperfect fit on a unit that must stay in: bias-corrected/augmented SC (ridge outcome model on the residuals; augsynth), or penalized SC.
• Many treated or disaggregated units: one SC per treated unit, aggregated; the penalized estimator (pensynth) restores uniqueness and sparsity for inside-the-hull units and spans pure SC to one-to-one matching; multisynth handles staggered timing.
• Both N and T modestly large: interactive fixed effects and matrix completion (gsynth, fect, MC-NNM) relax the convex-combination restriction entirely; the panel survey's unified view is that DiD, SC, unconfoundedness, and matrix completion are one imputation objective under different restrictions, so divergence across them is diagnostic.
• Volatile single-market outcomes with a Bayesian time-series counterfactual: CausalImpact, the tool marketing analytics teams usually already run for geo tests; the canon situates it among SC relatives, so treat its output with this skill's diagnostics, and note its counterfactual leans on one unit's own time series plus covariates.
• SC-type weights with many controls and few pre-periods need regularization; unregularized in-sample fit can be perfect and meaningless.
• Treated outcome outside the donor convex hull: augmented DiD (Li and Van den Bulte 2023), a different method from Ben-Michael's ridge-augmented SC in augsynth despite the near-identical name. Too few pre-periods: forward DiD (Li 2024); controls far fewer than pre-periods: HCW OLS (Hsiao, Ching, and Wan 2012). Fuller map in references/details.md.

R implementation

The complete runnable pipeline is scripts/synth_template.R (canonical SC with the full diagnostics battery, SDID, augmented and penalized variants, factor-model robustness, conformal and prediction intervals), with every call verified against package documentation. The core:

library(tidysynth)                       # canonical ADH workflow with built-in placebos
out <- df |>
  synthetic_control(outcome = y, unit = unit, time = year,
                    i_unit = "TREATED", i_time = T0, generate_placebos = TRUE) |>
  generate_predictor(time_window = pre_window, ...) |>
  generate_weights(optimization_window = pre_window) |>
  generate_control()
plot_trends(out); plot_placebos(out); grab_significance(out)   # RMSPE-ratio table

library(synthdid)                        # the SDID branch
setup <- panel.matrices(panel, unit = "unit", time = "year",
                        outcome = "y", treatment = "treated")
tau <- synthdid_estimate(setup$Y, setup$N0, setup$T0)
sqrt(vcov(tau, method = "placebo"))

Package index with versions, links, and traps in references/details.md.

Methods paragraph template

[Treatment] hit [treated unit] at [date]; no comparable unit did, so we construct a synthetic control from [J] donors, excluding [units] for [own interventions / shocks / spillovers] (Abadie 2021). Predictors are [pre-period outcomes and covariates]; predictor weights are chosen by [cross-validation on the pre-period], and the resulting synthetic unit puts weight on [named donors]. Pre-period fit is [shown in table/figure]. We report permutation inference with the post/pre RMSPE ratio over in-space placebos (Abadie, Diamond, and Hainmueller 2010); with [J] donors the smallest attainable p is 1/(J+1), [which is a limitation of the design, not the method: statistical significance in the conventional sense is out of reach and we lean on magnitude and the placebo distribution]. We validate with backdating, leave-one-out donor exclusion, and robustness across predictor sets and donor pools. [If fit is imperfect: because the synthetic unit cannot match pre-period levels exactly, we use synthetic difference-in-differences (Arkhangelsky et al. 2021), which differences out the remaining gap; I note this trades the perfect-fit requirement for the assumption that the gap would have been stable, which I cannot test directly.] The estimand is the effect on [treated unit] alone, and I generalize to [other units] only [not at all / under the stated assumption].

Every claim traces to references/canon.md; keys live in ../causal-design/references/causal.bib.

Handoffs

• did: parallel pre-trends hold, or staggered adoption with many treated units; synthetic DiD lives HERE, did points back for it.
• causal-design: whether any panel counterfactual is credible; the taxonomy that routes between did, SC, and factor models.
• rdd: policy-date designs masquerading as RD in time arrive here when one or a few aggregate units switch at a date; treat the date as the event, not a cutoff.
• field-experiment: prospective geo experiments (choosing treatment markets by design rather than analyzing one after the fact).
• preregister: locking weights and specification before post-period outcomes exist.