Methods: Synthetic Control • EventStudy

1. Title & Abstract

This article covers the synthetic control method (SCM) for estimating the effect of a single treated unit against a data-driven weighted combination of untreated “donor” units. SCM is the tool of choice when one unit is treated, many candidate controls exist, and no single control is a credible counterfactual on its own. It is estimated live here on inline synthetic treated-plus-donor data, explicitly choosing method = "optim" (pure base-R stats::optim; the package default is "quadprog" when that optional package is installed).

2. When to Use This Method

Use SCM for comparative case studies: one region, firm, or market receives a treatment (a law, a shock, an entry) and you need a counterfactual. Rather than pick one comparison unit, SCM forms a convex weighted average of donors that tracks the treated unit’s pre-treatment trajectory, then reads the post-treatment gap as the effect (Abadie et al. 2010).

3. Intuition

Blend the donor pool so the blend’s pre-treatment path lies on top of the treated unit’s path. If the synthetic unit and the real unit were indistinguishable before treatment, their divergence afterwards is the treatment effect. The blend weights are non-negative and sum to one, keeping the synthetic control inside the donors’ support (no extrapolation).

4. Model & Null Hypothesis

The synthetic counterfactual is a convex combination of donor outcomes, with weights chosen to minimise pre-period mean squared prediction error (MSPE):

\hat{Y}^{N}_{1t} = \sum_{j=2}^{J+1} w_j\, Y_{jt}, \qquad w_j \geq 0,\ \sum_j w_j = 1,

and the treatment effect is the post-treatment gap:

\text{gap}_t = Y_{1t} - \hat{Y}^{N}_{1t}.

The null hypothesis is H_0: \text{gap}_t = 0 for t \geq t_0 — the treated unit tracks its synthetic counterfactual after treatment, i.e. no effect.

5. Assumptions

Good pre-treatment fit: a low pre-period MSPE; a poor fit invalidates the counterfactual.
Convex hull: the treated unit’s pre-path lies within the donors’ span (weights on the simplex, no extrapolation).
No interference / spillovers: donors are unaffected by the treatment.
No anticipation before t_0.

6. Worked Example

We build inline synthetic data in base R with an explicit local set.seed(42): 5 donors over 80 periods (long format unit, time, outcome) and a treated unit (time, outcome) that tracks the donor average pre-treatment and jumps after t_0 = 60. We explicitly pass method = "optim" (pure stats::optim L-BFGS-B, no extra dependency); the package default is "quadprog" when that optional package is available.

library(EventStudy)
set.seed(42)

periods <- 1:80
ttime   <- 60
donor_ids <- paste0("D", 1:5)

donor_data <- do.call(rbind, lapply(donor_ids, function(d) {
  data.frame(unit = d, time = periods, outcome = cumsum(rnorm(80, 0.1, 1)))
}))

# Treated = donor-average pre-treatment + a post-treatment jump; lengths match 80.
donor_mean <- as.numeric(tapply(donor_data$outcome, donor_data$time, mean))
post_jump  <- ifelse(periods >= ttime, 0.15 * (periods - ttime + 1), 0)
treated_data <- data.frame(
  time    = periods,
  outcome = donor_mean + post_jump + rnorm(80, 0, 0.3)
)

task <- SyntheticControlTask$new(treated_data, donor_data, treatment_time = ttime)
res  <- estimate_synthetic_control(task, method = "optim")   # base R stats::optim

The quadprog quadratic-programming solver is available when that optional package is installed:

res_qp <- estimate_synthetic_control(task, method = "quadprog")

7. Rendered Table

knitr::kable(
  head(res$results$trajectory, 10),
  caption = "Treated vs synthetic trajectory with the estimated gap (first 10 periods)."
)

Treated vs synthetic trajectory with the estimated gap (first 10 periods).
time	treated	synthetic	gap
1	0.7993	0.3574	0.4419
2	0.1998	0.3995	-0.1996
3	0.9621	0.8875	0.0746
4	1.3397	1.2460	0.0937
5	1.2972	1.3703	-0.0731
6	1.9037	2.1786	-0.2748
7	2.7954	2.6887	0.1068
8	3.1292	2.8430	0.2862
9	4.0566	3.4895	0.5672
10	3.2637	3.7390	-0.4752

8. Rendered Plot

traj <- res$results$trajectory
plot(traj$time, traj$gap, type = "l",
     xlab = "Time", ylab = "Gap (treated - synthetic)",
     main = "Synthetic control gap")
abline(v = 60, lty = 2)
abline(h = 0, col = "grey60")

Gap between the treated unit and its synthetic control over time.

9. Interpretation

Before t_0 = 60 the gap should hover near zero — the synthetic control tracks the treated unit, confirming a credible counterfactual (low pre-period MSPE in res$results$pre_mspe). After t_0 the gap opens up, recovering the built-in post-treatment jump; its average is the ATT in res$results$att. A gap that was already non-zero before treatment would signal a poor fit and an untrustworthy estimate. Inference typically compares this treated-unit gap to the distribution of placebo gaps obtained by reassigning treatment to each donor.

References

Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. 2010. “Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program.” Journal of the American Statistical Association 105 (490): 493–505.