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Synthetic control method
Synthetic control method
Synthetic control method
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Comparison of per-capita GDP in West Germany before and after the 1990 German reunification and the hypothetical one if the reunification had not taken place.[1]

The synthetic control method is an econometric method used to evaluate the effect of large-scale interventions.[2] It was proposed in a series of articles by Alberto Abadie and his coauthors.[3][4][5] A synthetic control is a weighted average of several units (such as regions or companies) combined to recreate the trajectory that the outcome of a treated unit would have followed in the absence of the intervention. The weights are selected in a data-driven manner to ensure that the resulting synthetic control closely resembles the treated unit in terms of key predictors of the outcome variable.[3] Unlike difference in differences approaches, this method can account for the effects of confounders changing over time, by weighting the control group to better match the treatment group before the intervention.[6] Another advantage of the synthetic control method is that it allows researchers to systematically select comparison groups. It has been applied to the fields of economics,[7] political science,[1] health policy,[6] criminology,[8] and others.

The synthetic control method combines elements from matching and difference-in-differences techniques. Difference-in-differences methods are often-used policy evaluation tools that estimate the effect of an intervention at an aggregate level (e.g. state, country, age group etc.) by averaging over a set of unaffected units. Famous examples include studies of the employment effects of a raise in the minimum wage in New Jersey fast food restaurants by comparing them to fast food restaurants just across the border in Philadelphia that were unaffected by a minimum wage raise,[9] and studies that look at crime rates in southern cities to evaluate the impact of the Mariel Boatlift on crime.[10] The control group in this specific scenario can be interpreted as a weighted average, where some units effectively receive zero weight while others get an equal, non-zero weight.

The synthetic control method tries to offer a more systematic way to assign weights to the control group. It typically uses a relatively long time series of the outcome prior to the intervention and estimates weights in such a way that the control group mirrors the treatment group as closely as possible. In particular, assume we have J observations over T time periods where the relevant treatment occurs at time where Let

be the treatment effect for unit at time , where is the outcome in absence of the treatment. Without loss of generality, if unit 1 receives the relevant treatment, only is not observed for . We aim to estimate .

Imposing some structure

and assuming there exist some optimal weights such that

for , the synthetic controls approach suggests using these weights to estimate the counterfactual

for . So under some regularity conditions, such weights would provide estimators for the treatment effects of interest. In essence, the method uses the idea of matching and using the training data pre-intervention to set up the weights and hence a relevant control post-intervention.[3]

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophes and growth,[11] or civil conflicts and growth,[12] studies that examine the effect of vaccine mandates on childhood immunization,[13] and studies linking political murders to house prices.[14] Recently, the synthetic control method is actively used in new drug development when evaluating the causal impact of a treatment or intervention, especially in situations where randomized controlled trials (RCTs) are not feasible.[15]


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Synthetic control method

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The synthetic control method is a statistical approach for in comparative case studies, where a single treated unit—such as a state, , or —undergoes an intervention, and a synthetic counterfactual is formed by assigning non-negative weights to a set of untreated control units that optimally match the treated unit's pre-treatment outcome trajectory and relevant covariates. This weighting ensures the synthetic control approximates what the treated unit's outcome would have been absent the treatment, enabling estimation of the on the treated as the post-treatment gap between observed and synthetic outcomes. Developed by economists Alberto Abadie, Alexis Diamond, and Jens Hainmueller, the method builds on earlier work by Abadie and Javier Gardeazabal (2003) and was formalized in their 2010 paper in the Journal of the American Statistical Association, addressing gaps in traditional difference-in-differences estimators by minimizing researcher discretion in control selection and avoiding extrapolation beyond observed data through weights that sum to one and are bounded between zero and one. Unlike simple matching or averaging of controls, SCM optimizes for balance on multiple pre-treatment periods, providing transparent unit contributions and robust inference under factor model assumptions where unobserved confounders evolve similarly across units. The method's strengths include its applicability to rare events or single-unit interventions where randomized experiments are infeasible, its data-driven nature that reduces bias from comparisons, and its ability to handle heterogeneous treatment effects by focusing on transparent, reproducible counterfactuals. Early applications demonstrated its utility in estimating the reduction in cigarette sales from California's Proposition 99 tobacco control initiative (approximately 26 packs annually by 2000) and the short-term economic costs of on West Germany's GDP trajectory. Subsequent extensions, such as generalized and augmented variants, have refined inference and addressed limitations like sensitivity to pre-treatment fit or sparse donor pools, though the core approach remains prized for bridging qualitative insights with quantitative rigor in fields like , , and .

Overview

Definition and Purpose

The synthetic control method (SCM) is a statistical technique for estimating causal effects of interventions on aggregate outcomes in comparative case studies, especially when a single treated unit—such as a , state, or —lacks a direct untreated counterpart with parallel characteristics. Developed by economists Alberto Abadie and Javier Gardeazabal, SCM constructs a synthetic counterfactual by selecting optimal weights for a donor pool of untreated units to replicate the treated unit's pre-treatment values of the outcome variable and relevant predictors as closely as possible. This weighted combination, termed the synthetic control, approximates what the treated unit's trajectory would have been without the intervention, with the post-treatment gap between observed and synthetic outcomes serving as the effect estimate. The method's core purpose is to enable rigorous in non-experimental settings where traditional regression or matching approaches falter due to the absence of multiple treated units or violations of assumptions like parallel trends in difference-in-differences estimators. By prioritizing exact pre-treatment fit over post-treatment , SCM leverages longitudinal to isolate intervention effects while mitigating biases from unobserved confounders that vary over time but are stable across units in the pre-period. It has been applied to evaluate policies like laws, hikes, and economic shocks, providing transparent, replicable estimates grounded in observable rather than parametric modeling. Unlike propensity score methods, SCM avoids assuming functional forms for outcomes, emphasizing transparent weight optimization via minimization of in the pre-treatment phase.

Comparison to Difference-in-Differences and Other Methods

The synthetic control method (SCM) constructs a counterfactual for the treated unit by optimally weighting untreated donor units to replicate its pre-treatment outcome trajectory and relevant covariates, thereby avoiding the parallel trends assumption central to difference-in-differences (DiD) estimation. In DiD, treatment effects are identified under the assumption that treated and control groups would have followed parallel paths absent intervention, which can fail when units exhibit divergent pre-trends, leading to biased estimates if not addressed via extensions like fixed effects or trend controls. SCM mitigates this by design, as the synthetic control's weights ensure close matching on pre-treatment dynamics, making it preferable in applications with heterogeneous trends or where simple averaging of controls would distort the counterfactual. In the context of economic shocks affecting GDP, alternative methods for constructing counterfactual scenarios include extrapolating pre-event growth trends, such as assuming continued annual average growth of around 2% based on historical patterns, as commonly employed in vector autoregression (VAR)-based approaches. In contrast, SCM utilizes a "doppelgänger" approach by weighting similar economies to match the treated unit's pre-treatment characteristics, providing a more tailored counterfactual that better accounts for heterogeneous trends and post-event performance of donor units. SCM excels in scenarios with one or few treated units, such as evaluating nationwide policies, where DiD struggles due to limited and reliance on group-level averages that may not capture unit-specific paths. For instance, DiD requires multiple untreated units for robust averaging but assumes their collective trends proxy the treated unit's counterfactual, whereas SCM's (weights summing to 1 and non-negative) provides a tailored, data-driven surrogate without beyond donors. However, SCM demands extensive pre-treatment observations—typically 10 or more periods—and a pool of donors exceeding predictors to avoid , constraints less stringent in DiD, which can operate with shorter panels if parallel trends hold. Relative to matching methods, SCM extends exact or propensity score matching by incorporating time-series structure and multiple predictors, yielding a dynamic counterfactual rather than static pairs, though it shares sensitivity to donor pool quality and dimensionality. Unlike regression discontinuity designs (RDD), which identify local average treatment effects near a forcing variable cutoff via continuity assumptions, SCM applies to non-experimental, unit-level interventions without natural discontinuities, prioritizing global trajectory approximation over localized compliance. Compared to instrumental variables (IV) approaches, SCM eschews the need for exclusion restrictions and valid instruments, relying instead on observable pre-treatment data for identification, but it forgoes IV's ability to address endogeneity from unobserved confounders when donors imperfectly match latent factors. Inference in SCM typically employs placebo permutations on donors or time, contrasting DiD's standard errors or wild bootstraps, with recent hybrids like synthetic DiD blending both for improved precision in multi-unit settings.

History

Origins in Abadie and Gardeazabal (2003)

The synthetic control method was first proposed by economists Alberto Abadie and Javier Gardeazabal in their 2003 study examining the economic effects of associated with the Basque separatist group in Spain's Basque Country. Motivated by the challenge of estimating causal impacts in settings lacking directly comparable untreated units—such as other Spanish regions differentially affected by national policies or economic trends—the authors developed a data-driven approach to construct a counterfactual outcome trajectory for the treated unit. This method addressed limitations of traditional difference-in-differences estimators, which rely on parallel trends assumptions that may not hold when control groups are heterogeneous or influenced by unobserved confounders. In the Basque Country application, Abadie and Gardeazabal used annual per capita GDP data spanning 1955 to 1997, treating the intensification of terrorism—marked by a surge in attacks following the late 1960s and particularly after Francisco Franco's death in 1975—as the intervention. The synthetic control was formed as a weighted average of seven potential donor regions (excluding the Basque Country and Catalonia initially for matching), with weights optimized to minimize the difference between the Basque Country's pre-terrorism (primarily 1960s) characteristics and those of the synthetic counterpart. Key predictors for matching included per capita GDP, the investment-to-GDP ratio, and other economic indicators from the pre-intervention period, yielding weights dominated by Catalonia (0.8508) and Madrid (0.1492). This construction ensured the synthetic Basque Country closely replicated the actual region's trajectory before the conflict's escalation, assuming terrorism's effects were lagged and primarily negative on outcomes like GDP. Post-intervention comparisons revealed that Basque per capita GDP fell approximately 10 percentage points relative to the synthetic control after the 1970s, attributing this gap to terrorism's direct and indirect costs, such as reduced investment and . To assess robustness, the authors conducted a placebo test by applying the method to (which received the highest weight in the Basque synthetic control) as if it were treated, finding no significant divergence and even a 4% relative outperformance in Catalonia's GDP during 1990–1997, which suggested the Basque estimate might understate the true impact. Supplementary evidence from Basque stock returns during a 1998–1999 ETA truce showed a +10.14% abnormal gain, reversing to -11.21% after the truce ended, corroborating the method's ability to isolate conflict-related effects. This inaugural use established the synthetic control as a transparent, non-parametric tool for comparative case studies, emphasizing empirical matching over parametric modeling.

Extensions and Popularization (2010 Onward)

The synthetic control method gained substantial traction following its application in Abadie, Diamond, and Hainmueller's 2010 analysis of California's Proposition 99, a 1988 tobacco control program that increased cigarette taxes and anti- advertising; this study, published in the Journal of the , demonstrated the method's efficacy in estimating effects for single treated units by constructing a synthetic counterfactual from untreated states, matching pre-treatment rates and other predictors closely. The paper's emphasis on data-driven weights reduced researcher discretion compared to traditional case studies, fostering adoption in and for interventions lacking randomized controls, such as state-level reforms. By the mid-2010s, applications proliferated in areas like , labor , and , with over 500 citations of the 2010 work by 2020, reflecting its utility for aggregate treatments where parallel trends assumptions of difference-in-differences fail. Software implementations accelerated popularization, including the Synth package for R and Stata released around 2011, which operationalized the original estimator, and subsequent tools like gsynth for generalized variants. These packages enabled reproducible analyses of donor pools with dozens of units, standardizing placebo tests for inference and promoting the method's use in non-experimental settings, such as evaluating minimum wage hikes or immigration reforms. Guidance from Abadie et al. (2021) outlined feasibility criteria, like requiring at least 5-10 pre-treatment periods and sufficient donor variability, while cautioning against overfitting in small samples; this work highlighted SCM's strengths in transparent counterfactual construction but noted vulnerabilities to post-treatment donor contamination. Extensions post-2010 addressed limitations in handling multiple treated units, time-varying unobservables, and inference biases. Xu's 2017 generalized synthetic control method incorporated interactive fixed effects models, allowing estimation of average treatment effects under factor structures where unit-specific factors load differently over time, thus relaxing strict pre-treatment matching and extending to with heterogeneous effects; implemented in gsynth, it has been applied to multi-country shocks. Ben-Michael, Feller, and Rothstein's 2021 augmented synthetic control method refined estimation via ridge-penalized outcome models, improving pre-treatment fit and reducing bias from extrapolation outside the convex hull of donors, particularly when covariates are few or noisy; this approach enhances robustness in finite samples through cross-validation of penalties. A 2021 Journal of the American Statistical Association special section further advanced robust variants, linking SCM to principal components for and equivalence to estimators under low-rank assumptions. These developments widened SCM's scope to clustered treatments and high-dimensional settings, though empirical validity hinges on unconfounded donor selection and stable latent factors.

Methodology

Data Structure and Requirements

The synthetic control method requires comprising a single treated unit and a donor pool of multiple untreated s, observed across a sequence of time periods that include both pre-intervention and post-intervention phases. The core outcome variable, denoted YjtY_{jt}, must be recorded for the treated unit (j=1j=1) and each (j=2,…,J+1j=2, \dots, J+1), over TT total periods, with the intervention commencing after period T0T_0, such that pre-intervention data span t=1t=1 to T0T_0 and post-intervention data span t=T0+1t=T_0+1 to TT. This structure enables the construction of a counterfactual trajectory for the treated unit by weighting s to replicate its pre-intervention outcome path and associated characteristics. A sufficient length of pre-intervention periods (T0T_0) is essential for feasibility, as it allows for precise of weights that minimize discrepancies in the treated unit's predictors—typically including averages of the outcome variable over the pre-period or other time-invariant covariates unaffected by the intervention. The donor pool must contain enough units (often J≥10J \geq 10, though no strict minimum is mandated) to ensure the treated unit's pre-intervention features lie within or near the of the controls' features, facilitating a close synthetic match. Balanced panels are standard, with complete data on outcomes and predictors for all units in the pre-period; while basic implementations assume no missing values, extensions such as methods can accommodate imbalances or post-treatment gaps in controls. Data requirements also encompass the absence of anticipation effects in the treated unit prior to T0T_0 and adherence to the stable unit treatment value assumption, ensuring no spillovers or interference between units that could confound the control pool's validity. Predictor variables should be selected based on their to the outcome and immunity to treatment effects, with empirical practice emphasizing a parsimonious set to avoid while capturing structural similarities.

Constructing the Synthetic Control

The synthetic control is formed as a weighted average of untreated donor units, where the weights are selected to minimize the discrepancy between the treated unit's pre-intervention characteristics and those of the synthetic counterpart. This process relies on a pool of control units presumed unaffected by the intervention, typically chosen based on similarity in economic, institutional, or geographic features to the treated unit. The characteristics matrix X1X_1 for the treated unit includes averages of outcome variables over the pre-treatment periods (e.g., mean GDP from t=1t=1 to T0T_0) and additional predictors such as lagged outcomes, demographic factors, or sectoral shares that influence the outcome but are unaffected by the treatment. The corresponding matrix X0X_0 compiles these for the JJ donor units. Weights w∗=(w2∗,…,wJ+1∗)⊤w^* = (w_2^*, \dots, w_{J+1}^*)^\top are obtained by solving the optimization problem w∗=arg⁡min⁡w(X1−X0w)⊤V(X1−X0w)w^* = \arg\min_w (X_1 - X_0 w)^\top V (X_1 - X_0 w) subject to wj≥0w_j \geq 0 for all jj and ∑j=2J+1wj=1\sum_{j=2}^{J+1} w_j = 1, ensuring the synthetic control is a convex combination of donors. The symmetric positive definite diagonal matrix VV assigns relative weights to the predictors, emphasizing those with greater predictive power for the outcome; in early applications, VV was set to identity or simple diagonals, but subsequent refinements select its diagonal entries via cross-validation, splitting pre-treatment data (e.g., 1971–1980 for training, 1981–1990 for validation) to minimize root mean squared prediction error on held-out pre-period outcomes. The resulting weights yield a synthetic pre-treatment outcome ∑j=2J+1wj∗Yjt\sum_{j=2}^{J+1} w_j^* Y_{jt} that approximates Y1tY_{1t} closely for t≤T0t \leq T_0, often achieving near-perfect fit in aggregate applications like state-level policy evaluations. For instance, in estimating California's tobacco control program's effects, weights of 0.00 for most states but positive for others (e.g., higher for ) matched cigarette consumption and predictors like retail prices and incomes from –1987. Poor pre-treatment fit (e.g., large root mean squared prediction error) signals invalid donors or model misspecification, prompting exclusion of dissimilar units or addition of constraints for sparsity (e.g., limiting non-zero weights to 1–4 units). This matching condition underpins the counterfactual: post-treatment, the gap Y1t−∑j=2J+1wj∗YjtY_{1t} - \sum_{j=2}^{J+1} w_j^* Y_{jt} for t>T0t > T_0 estimates the on the treated, assuming the factor model generating the data—YitN=δt+θtZi+λtμi+εitY_{it}^N = \delta_t + \theta_t Z_i + \lambda_t \mu_i + \varepsilon_{it}, with unobserved factors λt\lambda_t and loadings μi\mu_i—permits via the weighted donors.

Estimation Procedure and Weights

The estimation procedure for synthetic control weights centers on solving a constrained optimization problem using pre-intervention data to construct a counterfactual that closely matches the treated unit's observed characteristics prior to treatment. Specifically, let unit 1 be the treated unit and units j=2,…,J+1j = 2, \dots, J+1 the potential controls from the donor pool; the vector of weights w∗=(w2∗,…,wJ+1∗)′\mathbf{w}^* = (w_2^*, \dots, w_{J+1}^*)' minimizes the quadratic form ∥X1−X0w∥V2=(X1−X0w)′V(X1−X0w)\| \mathbf{X}_1 - \mathbf{X}_0 \mathbf{w} \|_V^2 = (\mathbf{X}_1 - \mathbf{X}_0 \mathbf{w})' V (\mathbf{X}_1 - \mathbf{X}_0 \mathbf{w}), subject to wj≥0w_j \geq 0 for all jj and ∑j=2J+1wj=1\sum_{j=2}^{J+1} w_j = 1. Here, X1\mathbf{X}_1 is a K×1K \times 1 vector of pre-treatment predictors for the treated unit, including time-invariant covariates Z1\mathbf{Z}_1 and transformations of pre-treatment outcomes such as averages Yˉ1=T0−1∑t=1T0Y1t\bar{Y}_{1} = T_0^{-1} \sum_{t=1}^{T_0} Y_{1t}; X0\mathbf{X}_0 is the analogous K×JK \times J matrix for controls; and VV is a K×KK \times K symmetric positive semidefinite diagonal matrix that weights the relative importance of each predictor, often calibrated to minimize mean squared prediction error (MSPE) on held-out pre-treatment periods. This optimization yields a of controls, ensuring the synthetic unit lies within the of the donor pool and avoiding beyond observed data points, which enhances transparency and interpretability relative to methods permitting negative weights. The problem is solved via algorithms, as implemented in packages like Synth in , which iteratively adjust weights to achieve the minimum discrepancy. In the original application to the Basque Country's terrorism costs, Abadie and Gardeazabal (2003) specified predictors including initial per capita GDP, population shares, and sectoral investments, selecting VV's elements to best replicate the Basque GDP trajectory in the 1960s, resulting in nonzero weights primarily for (85%) and (15%). Post-treatment, for t>T0t > T_0, the counterfactual outcome is Y^1tN=∑j=2J+1wj∗Yjt\hat{Y}_{1t}^N = \sum_{j=2}^{J+1} w_j^* Y_{jt}, and the treatment effect estimate is α^1t=Y1t−Y^1tN\hat{\alpha}_{1t} = Y_{1t} - \hat{Y}_{1t}^N. Extensions refine this core procedure; for instance, Abadie (2021) recommends splitting pre-intervention periods into training and validation subsets to select V∗V^* via out-of-sample MSPE minimization, reducing risks when predictor dimensionality approaches the number of pre-treatment periods. While the standard approach enforces non-negativity for causal realism under parallel trends-like assumptions, alternatives like Doudchenko and Imbens (2016) relax this via elastic net penalization, allowing negative weights but introducing potential biases. The number of nonzero weights is typically bounded by the number of predictors KK, promoting sparsity and economic interpretability in the synthetic composition.

Assumptions and Identification Strategy

Core Identifying Assumptions

The synthetic control method identifies causal effects under a linear interactive fixed effects framework for untreated potential outcomes. For control units j=2,…,Jj = 2, \dots, J and all time periods tt, the outcome is YjtN=δt+θt′Zj+λt′μj+εjtY_{jt}^N = \delta_t + \theta_t' Z_j + \lambda_t' \mu_j + \varepsilon_{jt}, where δt\delta_t captures common time shocks, ZjZ_j denotes observed time-invariant covariates, μj\mu_j represents unobserved unit-specific factors, λt\lambda_t are unobserved time-varying factor loadings, and εjt\varepsilon_{jt} is a zero-mean idiosyncratic error orthogonal to the factors and covariates. This model assumes that systematic variation in outcomes arises from these interactive fixed effects, with transitory shocks εjt\varepsilon_{jt} being small relative to the signal, particularly in pre-treatment periods, to enable precise matching. Identification hinges on constructing non-negative weights w∗=(w2∗,…,wJ∗)w^* = (w_2^*, \dots, w_J^*) summing to one that minimize the discrepancy between the treated unit's pre-treatment outcomes Y1tY_{1t} for t≤T0t \leq T_0 and the synthetic counterpart ∑j=2Jwj∗Yjt\sum_{j=2}^J w_j^* Y_{jt}, often incorporating covariates Z1Z_1. This matching implies that the synthetic control approximates the treated unit's unobserved factors μ1\mu_1 and covariates Z1Z_1, such that absent treatment, post-treatment outcomes would have evolved similarly: Y1tN≈∑j=2Jwj∗YjtY_{1t}^N \approx \sum_{j=2}^J w_j^* Y_{jt} for t>T0t > T_0. The assumption requires that the donor pool spans the support of the treated unit's factor loadings, allowing exact or near-exact replication of pre-trends without relying on parallel trends across raw controls. Crucial auxiliary assumptions include the absence of spillovers, ensuring treatment affects only the treated unit and not controls, and no , whereby effects manifest only after T0T_0. Additionally, the framework presumes stable causal mechanisms post-treatment, meaning unobserved confounders captured by μi\mu_i and λt\lambda_t do not evolve differentially between the treated and synthetic units due to the intervention. Violations, such as time-varying unobserved confounders uncorrelated with pre-treatment data, can bias estimates, underscoring the need for substantive knowledge to validate the approximation.

Inference via Placebo and Permutation Tests

In the synthetic control method (SCM), standard parametric inference procedures are often unsuitable due to the typical presence of a single treated unit and a small number of donor pool controls, which precludes asymptotic approximations and variance estimation under conventional assumptions. Instead, Abadie, Diamond, and Hainmueller (2010) propose nonparametric inference based on placebo and permutation tests, which generate an empirical distribution of test statistics under a sharp null hypothesis of no treatment effect to assess significance. These approaches leverage the structure of comparative case studies by simulating "placebo interventions" or random reassignments, providing exact finite-sample inference without relying on large-sample normality. Placebo tests form the core of this inference framework, involving the application of the SCM to each untreated unit in the donor pool as if it had received the intervention. For a given placebo unit jj, a synthetic control is constructed using the remaining donor units (excluding jj) to match pre-intervention outcomes and covariates, yielding a counterfactual . The post-intervention discrepancy—typically measured as the root mean squared prediction error (RMSPE), defined as 1T−T0∑t>T0(Yjt−Y^jtN)2\sqrt{\frac{1}{T - T_0} \sum_{t > T_0} (Y_{jt} - \hat{Y}_{jt}^N)^2}
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