Inclusive Synthetic Control Method

• Inclusive Synthetic Control Method (iSCM) is a refined synthetic control approach that retains potentially affected donor units and corrects for spillover contamination.
• It employs standard pre-treatment synthetic control optimization to estimate weights and then solves a linear system to separate treatment from spillover effects.
• Empirical simulations and applications demonstrate that iSCM improves pre-treatment fit and reduces bias compared to standard and restricted SCM methods.

Inclusive Synthetic Control Method (iSCM) is a modification of synthetic control methods for settings in which units in the donor pool may be directly or indirectly affected by the intervention, so that SUTVA is violated and standard donor exclusion can materially worsen pre-treatment fit. In its 2024 formulation, iSCM retains “potentially affected” units in the donor pool, estimates synthetic control weights using only pre-intervention data, and then removes post-intervention contamination by solving a linear system that separates the treatment effect on the main treated unit from spillover effects on affected donors (Stefano et al., 2024). A later comparative study describes the same logic operationally as estimating standard SCM weights for all included units, computing post-treatment synthetic-control gaps, and recovering latent effects through an algebraic contamination correction (Melnychuk, 2024).

1. Problem setting and estimands

The canonical setup indexes units by $j = 1, \ldots, J$ and time by $t = 1, \ldots, T$, with an intervention at time $T_0$. Unit $1$ is the “main treated” unit. Within the donor pool, $m < J-2$ units, indexed $2,\ldots,m$, are “potentially affected”: they may be directly treated or indirectly affected by spillovers. Units $m+1,\ldots,J$ are “pure controls” (Stefano et al., 2024).

The potential-outcomes notation distinguishes three post-intervention states. $Y_{1t}^I$ denotes the outcome of the main treated unit under intervention; $Y_{jt}^S$, $j=2,\ldots,m$, denotes the outcome of potentially affected units under spillover or treatment; and $t = 1, \ldots, T$0 denotes the outcome in the absence of intervention. The effects of interest are

$t = 1, \ldots, T$1

The original iSCM formulation adopts a partial-interference structure with a known affected set $t = 1, \ldots, T$2. Assumption 1 states that in the pre-intervention period $t = 1, \ldots, T$3 for all units; in the post-intervention period $t = 1, \ldots, T$4 for pure controls, $t = 1, \ldots, T$5 for the main treated unit, and $t = 1, \ldots, T$6 for potentially affected units (Stefano et al., 2024). In this formulation, iSCM does not formalize adjacency matrices, network exposure mappings, or parametric spillover intensities. Instead, the analyst specifies ex ante which units may be affected and then treats spillovers as unit-level post-treatment effects on those units.

This construction is motivated by a specific failure mode of standard SCM. If affected donors are excluded, the donor pool may shrink enough to degrade predictor matching and RMSPE; if they are retained without correction, post-treatment synthetic outcomes are contaminated by donors’ own treatment or spillover effects. iSCM targets precisely this tradeoff.

2. Estimator construction and contamination correction

iSCM preserves the pre-treatment fitting stage of SCM. For the main treated unit, one estimates synthetic-control weights using a standard SCM-type estimator on pre-treatment data only. A generic formulation is

$t = 1, \ldots, T$7

or, in predictor form,

$t = 1, \ldots, T$8

No change to this optimization is required by iSCM (Melnychuk, 2024).

The distinctive step is post-estimation. For the main treated unit, the standard SCM counterfactual is

$t = 1, \ldots, T$9

and the usual post-treatment gap is $T_0$0. Because affected donors enter with observed post-treatment outcomes $T_0$1, this gap includes contamination terms of the form $T_0$2. iSCM therefore constructs synthetic controls not only for the main treated unit but also for each potentially affected unit $T_0$3, using donor pools that may include the main treated unit, other affected units, and pure controls. This produces biased post-treatment gaps $T_0$4 for each affected unit (Stefano et al., 2024).

Stacking these biased gaps yields the linear system

$T_0$5

with estimator

$T_0$6

The same logic is expressed more compactly in the comparative study as

$T_0$7

where $T_0$8 is the vector of observed synthetic-control gaps, $T_0$9 is the vector of latent treatment and spillover effects, and $1$0 is the matrix of pre-treatment synthetic-control weights (Melnychuk, 2024).

In this sense, “inclusive” means that spillover-affected donors remain in the donor pool and their contamination is removed analytically rather than excluded ex ante. The correction is period-by-period when there are multiple post-treatment periods.

For the two-unit affected case $1$1, the system has a closed form: $1$2 This special case makes explicit that iSCM is a bias-correction layer applied after standard SCM weights have been estimated.

3. Identification conditions and theoretical properties

The original iSCM paper states four core assumptions. Assumption 2 requires that, as the number of pre-intervention periods $1$3 goes to infinity, the SCM-type estimator used for the main treated unit satisfies

$1$4

Assumption 3 requires that, as $1$5, the SCM estimator used for each potentially affected unit $1$6 satisfies

$1$7

Assumption 4 requires that $1$8 is non-singular (Stefano et al., 2024).

The comparative study restates the same identifying logic in operational terms. The relevant conditions are stable weights across time, additive interference structure, sufficient pre-treatment fit, and invertibility of $1$9. Under additive effects,

$m < J-2$0

and with sufficiently good pre-treatment fit, the post-treatment gap for unit $m < J-2$1 reflects its own latent effect minus the weighted contamination transmitted through donors’ effects (Melnychuk, 2024).

The main theoretical result in the original formulation is Theorem 1: $m < J-2$2 This yields consistency of the treatment and spillover effect estimates under Assumptions 1–4 (Stefano et al., 2024).

The bias comparison in the two-unit affected case clarifies what iSCM removes. Standard SCM bias for the main treated unit contains a spillover term,

$m < J-2$3

whereas the iSCM bias becomes

$m < J-2$4

The implication is specific: standard SCM remains biased whenever an affected donor receives positive weight, even under perfect pre-treatment fit, while iSCM’s asymptotic bias vanishes as the underlying SCM approximation errors vanish (Stefano et al., 2024).

Invertibility is a substantive condition rather than a mere algebraic convenience. The paper notes that $m < J-2$5 is singular only in extreme cases, such as two units assigning weight $m < J-2$6 to each other and all pure-control weights being zero. In the comparative formulation, near-singularity of $m < J-2$7 can destabilize recovered effects, which motivates condition-number checks and regularization (Melnychuk, 2024).

4. Computation, diagnostics, and inference

The implementation is modular. Step 1 estimates pre-treatment SCM weights for each included unit. Step 2 computes post-treatment synthetic-control gaps. Step 3 solves the contamination system. In the comparative paper, the algorithm is presented as: estimate $m < J-2$8, form $m < J-2$9, and solve

$2,\ldots,m$0

for each post-treatment period. If donor sets differ across units, $2,\ldots,m$1 is filled with zeros where $2,\ldots,m$2 (Melnychuk, 2024).

Computationally, iSCM requires one SCM fit for the main treated unit and one for each potentially affected unit. The matrix inversion is low-dimensional in the applications for which SCM is typically used. The comparative study gives the linear solve as $2,\ldots,m$3 per post-treatment period, which is negligible for small donor pools typical in SCM applications (Melnychuk, 2024).

Diagnostics are central because iSCM delegates identification to the quality of pre-treatment weight estimation. The comparative study recommends validating pre-treatment fit for all included units via MSPE pre-treatment and goodness-of-fit plots, checking the conditioning of $2,\ldots,m$4, and using ridge stabilization when the system is ill-conditioned: $2,\ldots,m$5 with small $2,\ldots,m$6, together with reported sensitivity to $2,\ldots,m$7 (Melnychuk, 2024).

The original iSCM paper proposes placebo-in-space tests adapted to the contamination setting. For the main treated effect, estimated spillovers are subtracted from affected donors in post-treatment periods before computing the post/pre RMSPE ratio $2,\ldots,m$8; for each affected unit, all other estimated spillovers and the estimated main treatment effect are removed from donor outcomes before constructing the corresponding RMSPE ratio $2,\ldots,m$9 (Stefano et al., 2024). The paper also notes that conformal, subsampling, and bootstrap-based procedures for SCM can be adapted by accounting for post-treatment contamination when constructing null datasets and residuals.

A practical constraint is software access to weights. The comparative study notes explicitly that iSCM requires access to synthetic-control weights for included units; if weights are not available from the SCM implementation, iSCM cannot be applied directly. A repository containing implementations for Unrestricted, Restricted, iSCM, SP SCM, and Iterative SCM used in the simulations is provided at https://github.com/Melnychuk-Andrii/Spillover-SCM (Melnychuk, 2024).

5. Comparative performance and empirical illustrations

The main comparative evidence comes from Monte Carlo simulations based on factor-model data-generating processes following Cao and Dowd (2019), in both Stationary and $m+1,\ldots,J$0 cases. The simulation grid crosses pre-treatment period length $m+1,\ldots,J$1 or $m+1,\ldots,J$2, number of control units $m+1,\ldots,J$3 or $m+1,\ldots,J$4, ratio of spillover units $m+1,\ldots,J$5, $m+1,\ldots,J$6, $m+1,\ldots,J$7, or $m+1,\ldots,J$8, treatment effect $m+1,\ldots,J$9 or $Y_{1t}^I$0, ratio spillover/treatment $Y_{1t}^I$1, $Y_{1t}^I$2, $Y_{1t}^I$3, or $Y_{1t}^I$4, and a single post-treatment period. For each of $Y_{1t}^I$5 parameter combinations, $Y_{1t}^I$6 datasets were generated, producing $Y_{1t}^I$7 MSPE points across five methods (Melnychuk, 2024).

Method	Stationary MSPE	I(1) MSPE
iSCM	2.4233	2.5362
Iterative SCM	2.4735	2.5634
SP SCM	2.6225	2.7534
Restricted SCM	3.9856	2.5899
Unrestricted SCM	3.5666	3.5557

These results support a specific empirical ranking: iSCM performs best overall in both DGP classes, with Iterative SCM a close second. The comparative paper further reports that as the ratio of spillover units increases, Unrestricted SCM degrades; for very high ratios such as $Y_{1t}^I$8, Restricted SCM performs poorly in the Stationary case, and iSCM, Iterative, and SP SCM converge to similar MSPEs, with iSCM typically lowest. Larger treatment effects and proportionally larger spillovers increase Unrestricted SCM bias. In the $Y_{1t}^I$9 case, iSCM, Iterative, SP SCM, and Restricted slightly degrade with longer pre-treatment length, while Unrestricted improves marginally; in the Stationary case, most methods improve slightly with longer pre-treatment, except Restricted SCM (Melnychuk, 2024).

The original iSCM paper’s empirical illustration revisits German reunification. The panel covers OECD countries from 1960 to 2003, with intervention at 1990. West Germany is the main treated unit and Austria is designated as potentially affected because of trade and investment diversion. In unrestricted SCM, Austria receives weight $Y_{jt}^S$0 in synthetic West Germany; in synthetic Austria, West Germany receives weight $Y_{jt}^S$1. The unrestricted SCM RMSPE for West Germany is $Y_{jt}^S$2, versus $Y_{jt}^S$3 for restricted SCM. In the two-unit affected case, $Y_{jt}^S$4, and the resulting iSCM estimates imply that the negative post-reunification effect on West Germany’s GDP per capita is slightly larger in magnitude—up to approximately $Y_{jt}^S$5 more negative—than standard SCM, while Austria exhibits a small negative spillover of up to approximately $Y_{jt}^S$6 USD that is likely not statistically significant by placebo evidence (Stefano et al., 2024).

The comparative paper also revisits California Tobacco Control, West Germany Reunification, Basque Country Terrorism, and the 2014 Eastern Ukrainian Crisis. In the “No covariates” setting, iSCM lines are reported alongside other methods; with covariates, iSCM and SP SCM were not shown because of implementation issues involving weights access and covariate handling. Across examples, inclusive approaches align with or improve over Unrestricted SCM, especially where spillovers are plausible (Melnychuk, 2024).

6. Relation to alternative SCM variants

The immediate benchmark is baseline SCM with all donors retained and no correction. In that setting, the post-treatment gap conflates the true effect with contamination transmitted through affected donors’ weights. Restricted SCM removes suspected spillover donors ex ante in an attempt to restore SUTVA, but this can substantially worsen fit when the donor pool is small or many controls are affected. iSCM differs by retaining all donors and correcting contamination analytically. Iterative SCM, introduced in the comparative study, instead constructs synthetic versions of affected donors using only clean donors and then uses those synthetic donors in a final SCM run for the treated unit; unlike iSCM, it does not require access to SCM weights for all units and does not solve a single joint system. SP SCM incorporates a parametric linear structure for spillovers and performs joint estimation and testing under that structure, whereas iSCM does not impose a parametric spillover model (Melnychuk, 2024).

A recurrent misconception is to equate iSCM with any “inclusive” weighting strategy. The 2025 paper on $Y_{jt}^S$7-regularized synthetic control explicitly states that its estimator is not labeled “iSCM” and does not target the same problem as Di Stefano and Mellace (2024). Its inclusivity is instead a weighting philosophy: penalizing the maximum donor weight to obtain denser, more evenly distributed donor weights in a valid donor pool without spillovers (Wang et al., 30 Oct 2025). The substantive similarity is therefore limited to donor-weight dispersion rather than spillover accommodation.

The term “inclusive” is also used in a different sense in the Bayesian maximum-a-posteriori paper on synthetic control with convex hull restrictions. There, the inclusive construction is a “parallelly shiftable convex hull” that allows a vertical/intercept shift $Y_{jt}^S$8 while retaining nonnegative weights summing to one: $Y_{jt}^S$9 The fitting problem is

$j=2,\ldots,m$0

which addresses permanent additive differences rather than spillover contamination (Goh et al., 2020). The shared label reflects inclusiveness in feasible matching geometry, not the interference-oriented system inversion that defines the 2024 iSCM.

7. Limitations, edge cases, and open questions

iSCM requires a priori identification of the set of “potentially affected” units. If spillovers are pervasive or the affected set is misspecified, Assumption 1 may fail. The original paper also states that iSCM needs at least one pure control with nonzero weight; as $j=2,\ldots,m$1 grows, finite-sample reliability declines, the contamination system may become near-singular, and estimation errors can be amplified (Stefano et al., 2024).

The comparative study identifies several unresolved issues. Covariates were not used in the artificial datasets, even though in empirical practice covariates often improve pre-fit and may affect iSCM performance. Only two DGPs—Stationary and $j=2,\ldots,m$2—were tested. Spillover unit misspecification, including under- and over-specification, was not explored. Placebos and robustness checks were not implemented in the simulations, despite being recommended for empirical credibility. Computational limits constrained $j=2,\ldots,m$3, $j=2,\ldots,m$4, and iteration counts, and iSCM and SP SCM had difficulties with covariates in empirical examples (Melnychuk, 2024).

Several practical implications follow directly from these limitations. iSCM is most attractive when excluding potentially affected donors materially worsens pre-treatment fit, when a substantial share of controls may be spillover-affected, and when synthetic-control weights are available for all included units. The comparative paper states that if many donors may be spillover-affected—more than approximately $j=2,\ldots,m$5 of controls—iSCM avoids shrinking the donor pool and generally performs well; if fewer than approximately $j=2,\ldots,m$6 of controls are affected, Restricted SCM often performs similarly with simpler implementation (Melnychuk, 2024). This suggests that iSCM is best viewed not as a universal replacement for SCM, but as a targeted correction for small-$j=2,\ldots,m$7, long-$j=2,\ldots,m$8 settings in which donor contamination is plausible and donor exclusion is costly.