Synthetic Control

Motivating Example

In the early 2000s, researchers wanted to know: what was the economic cost of terrorism in the Basque Country? ETA's campaign of political violence had been raging since the late 1960s. Could you estimate the causal effect of terrorism on GDP?

The challenge is immediate and severe. There is only one Basque Country. You cannot randomly assign terrorism to some regions and not others. Standard DiD requires a comparable control group, but no single Spanish region is a convincing stand-in for the Basque Country — the region has a unique industrial structure, culture, and geography.

Abadie and Gardeazabal (2003) proposed an innovative solution: what if you could construct a synthetic Basque Country — a weighted combination of other Spanish regions whose economic trajectory before terrorism closely matched the real Basque Country? If this synthetic version tracked the Basque Country closely before terrorism began, then the gap between the real and synthetic Basque Country after terrorism is a credible estimate of the causal effect.

This paper introduced the synthetic control method — a widely adopted methodological development in modern causal inference.

A. Overview

The synthetic control method constructs a counterfactual for a single treated unit by finding a weighted average of untreated ("donor") units that best resembles the treated unit in the pre-treatment period. The key insight is that rather than choosing one comparison unit, you can build a better comparison by combining several.

The method was formalized and extended by Abadie et al. (2010) in their study of California's Proposition 99, an aggressive tobacco control program.

They showed that California's per-capita cigarette sales dropped dramatically after Proposition 99 relative to a synthetic California constructed from a weighted combination of other states.

When to Use Synthetic Control

The method shines when:

• You have one or a few treated units at an aggregate level (country, state, industry)
• You have a rich pre-treatment period with many time points
• You have a pool of donor units that were not affected by the treatment
• No single donor unit is a convincing control on its own

It is less appropriate when:

• You have many treated units (use DiD or staggered DiD instead)
• The pre-treatment period is very short
• The donor pool is small or contaminated by spillovers

Common Confusions

"Is synthetic control just matching?" It is related to matching, but there are important differences. Matching typically operates at the individual level, matching on pre-treatment covariates. Synthetic control operates at the aggregate level, matching on the pre-treatment trajectory of the outcome variable itself. Matching on outcomes rather than just covariates is a stronger form of matching.

"What if the synthetic control does not fit well in the pre-treatment period?" Then the method is not appropriate for your setting. A poor pre-treatment fit means the convex combination of donors cannot reproduce the treated unit's trajectory, which undermines the entire logic. If the fit is poor, this failure is an honest signal that the method does not work for your setting — not a problem to fix by adjusting weights.

"Can I include the treated unit's own pre-treatment outcome as a predictor?" Yes, and doing so is standard practice. Abadie et al. (2010) recommend including lagged outcomes as predictors alongside other covariates. Including the full set of pre-treatment outcome lags effectively means the synthetic control is chosen to match the treated unit's entire pre-treatment trajectory.

B. Identification

Setup

Let $Y_{1t}$ be the outcome for the treated unit at time $t$, and let $Y_{jt}$ for $j = 2, \ldots, J+1$ be outcomes for the $J$ donor units. Treatment occurs at time $T_0 + 1$.

The synthetic control is a vector of weights $\mathbf{W} = (w_2, \ldots, w_{J+1})'$ such that:

• $w_j \geq 0$ for all $j$ (non-negativity)
• $\sum_j w_j = 1$ (weights sum to one)

The synthetic control outcome is: $\hat{Y}_{1t}^{SC} = \sum_{j=2}^{J+1} w_j^* Y_{jt}$

The estimated treatment effect at time $t > T_0$ is:

Choosing the Weights

The weights $\mathbf{W}^*$ are chosen to minimize the discrepancy between the treated unit and the synthetic control in the pre-treatment period:

where $\mathbf{X}_1$ is a vector of pre-treatment characteristics (including lagged outcomes) for the treated unit, $\mathbf{X}_0$ is the corresponding matrix for donor units, and $V$ is a positive semidefinite matrix that weights the importance of different predictors.

Identifying Assumption

The key assumption is that the synthetic control constructed from the pre-treatment period remains a valid counterfactual in the post-treatment period. Abadie et al. (2010) show that if the treated unit's outcome is generated by a linear factor model:

then a synthetic control that closely matches the treated unit's pre-treatment outcomes will also match the unobserved factors $\boldsymbol{\mu}_i$, provided the number of pre-treatment periods is large enough.

This factor model is weaker than parallel trends in that it allows for heterogeneous trends — it allows for unit-specific trends driven by the factor loadings $\boldsymbol{\mu}_i$. But it is stronger in another way: it requires a good pre-treatment fit, which may not always be achievable.

C. Visual Intuition

The classic synthetic control presentation has two panels:

Panel A: Trajectory plot. The treated unit's outcome over time (solid line) alongside the synthetic control (dashed line). Before treatment, they should overlap closely. After treatment, any gap is the estimated effect.

Panel B: Gap plot. The difference $Y_{1t} - \hat{Y}_{1t}^{SC}$ over time. Before treatment, this should hover around zero. After treatment, persistent deviations indicate a treatment effect.

D. Mathematical Derivation

E. Implementation

1# Using the Synth package (original implementation)
2library(Synth)
3
4# Step 1: Prepare the data for synthetic control
5# foo = dataset, predictors = covariates to match on
6# dependent = outcome variable name
7# treatment.identifier = ID of the treated unit
8# controls.identifier = IDs of donor (control) units
9# time.predictors.prior = pre-treatment years for predictor matching
10# time.optimize.ssr = pre-treatment years for outcome matching
11dataprep_out <- dataprep(
12foo = df,
13predictors = c("pred1", "pred2"),
14predictors.op = "mean",
15dependent = "outcome",
16unit.variable = "unit_id",
17time.variable = "year",
18treatment.identifier = treated_id,
19controls.identifier = control_ids,
20time.predictors.prior = 1980:1993,
21time.optimize.ssr = 1980:1993,
22time.plot = 1980:2005
23)
24
25# Step 2: Run the synthetic control optimization
26synth_out <- synth(dataprep_out)
27
28# Step 3: Plot treated vs. synthetic control trajectories
29path.plot(synth_out, dataprep_out, Main = "Treated vs. Synthetic Control")
30
31# Step 4: Plot the gap (treatment effect over time)
32gaps.plot(synth_out, dataprep_out, Main = "Treatment Effect (Gap)")
33
34# Alternative: Using augsynth (recommended for modern features)
35# Provides bias correction, ridge regularization, and automatic inference
36library(augsynth)
37syn <- augsynth(outcome ~ treatment, unit = unit_id, time = year, data = df)
38summary(syn)
39plot(syn)

F. Diagnostics

1. Pre-treatment fit. An essential diagnostic. If the synthetic control does not closely track the treated unit before treatment, the method fails. Report the root mean squared prediction error (RMSPE) for the pre-treatment period.

2. Donor weights. Report which units get positive weight and how much. If the synthetic control concentrates weight on a single donor, you are effectively doing a single-unit comparison. If weights are spread across many units, the synthetic control is a genuine average.

3. Predictor balance. Report a table comparing the treated unit, the synthetic control, and the unweighted average of all donors on all predictor variables. The synthetic control should be much closer to the treated unit than the simple average.

4. In-space placebos. Run the synthetic control analysis for every donor unit (pretending each one was treated). If the treated unit's gap is unusually large relative to the placebo gaps, this evidence supports a genuine treatment effect. This procedure provides a permutation-style p-value.

5. In-time placebos. Re-run the analysis using a fake treatment date in the pre-treatment period. If you see a "gap" at the fake treatment date, something is wrong with the specification.

6. Leave-one-out robustness. Re-estimate removing each donor unit one at a time. If the results change dramatically when one donor is removed, the synthetic control is fragile.

Interpreting Your Results

Good pre-treatment fit, large post-treatment gap: This pattern is the ideal result. The synthetic control tracks the treated unit well before treatment, and the post-treatment divergence is your estimated effect. Bolster this with placebo tests.

Poor pre-treatment fit: Do not proceed. A synthetic control that cannot match the pre-treatment trajectory is not a credible counterfactual. Consider a different method or a different research question.

Effect appears immediately and persists: Consistent with a permanent treatment effect.

Effect grows over time: The treatment may take time to fully materialize.

Effect appears and then fades: The treatment effect may be temporary.

G. What Can Go Wrong

H. Practice

Placebo p-value: 2/18 = 0.11

I. Swap-In: When to Use Something Else

• Difference-in-differences: When many units are treated and parallel trends is defensible — DiD is simpler and provides standard inference.
• Synthetic DiD: When you want to combine the unit-reweighting logic of synthetic control with the time-differencing logic of DiD, especially with multiple treated units.
• Event studies: When the full time profile of treatment effects is of primary interest and many treated units are available.
• Matching: When many treated units exist and covariate-based matching on pre-treatment characteristics is feasible.

J. Reviewer Checklist