Synthetic Control Methods

Simon Mueller

2026-10-09

Introduction

Synthetic control methods (Abadie and Gardeazabal, 2003; Abadie, Diamond, and Hainmueller, 2010) provide a data-driven approach to construct a counterfactual for a single treated unit from a weighted combination of donor (control) units. This is particularly useful when:

The EventStudy package implements synthetic control estimation with two optimization methods and includes a placebo test for inference.

Data Setup

Synthetic control requires: - Treated unit: time series of the outcome for the treated unit - Donor pool: time series of outcomes for potential control units - Treatment time: when the intervention occurs

library(EventStudy)
library(tibble)

# Treated unit
treated_data <- tibble(
  time = 1:100,
  outcome = cumsum(rnorm(100, mean = 0.1, sd = 0.5))
)
# Inject treatment effect after period 60
treated_data$outcome[61:100] <- treated_data$outcome[61:100] + 5

# Donor pool (long format)
donor_data <- do.call(rbind, lapply(paste0("D", 1:10), function(u) {
  tibble(
    unit = u,
    time = 1:100,
    outcome = cumsum(rnorm(100, mean = 0.1, sd = 0.5))
  )
}))

Creating the Task

task <- SyntheticControlTask$new(
  treated_data = treated_data,
  donor_data = donor_data,
  treatment_time = 61
)
print(task)
#> SyntheticControlTask
#>   Donors:      10
#>   Periods:     100
#>   Pre-treat:   60
#>   Post-treat:  40
#>   Treatment:   61
#>   Estimated:   FALSE

Estimation

Two optimization methods are available:

quadprog (default)

Uses quadprog::solve.QP() for exact quadratic programming. Requires the quadprog package.

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

optim (fallback)

Uses stats::optim() with a softmax reparameterization. No extra dependencies.

task <- estimate_synthetic_control(task, method = "optim")

Results

After estimation, the task contains:

# Donor weights (sum to 1, all >= 0)
task$results$weights
#>     D1     D2     D3     D4     D5 ...
#>  0.312  0.000  0.188  0.000  0.500 ...

# Average treatment effect on the treated
task$results$att
#> [1] 4.87

# Pre- and post-treatment mean squared prediction error
task$results$pre_mspe
task$results$post_mspe

# Full trajectory
task$results$trajectory
#> # A tibble: 100 x 4
#>     time treated synthetic   gap
#>    <int>   <dbl>     <dbl> <dbl>
#>  1     1   0.234     0.289 -0.0547
#>  ...

Visualization

Three plot types are available:

Trajectory Plot

Compare the treated unit to its synthetic counterpart:

plot_synthetic_control(task, type = "trajectory")

Gap Plot

Visualize the treatment effect (gap between treated and synthetic):

plot_synthetic_control(task, type = "gap")

Placebo Plot

After running the placebo test, overlay placebo gaps:

task <- sc_placebo_test(task)
plot_synthetic_control(task, type = "placebo")

Placebo Test

The placebo test re-estimates the synthetic control for each donor unit as a pseudo-treated unit. The treated unit’s RMSPE ratio is ranked among all units to produce a p-value:

\[p = \frac{\text{rank of treated ratio}}{N_{donors} + 1}\]

where the RMSPE ratio is \(\frac{RMSPE_{post}}{RMSPE_{pre}}\).

task <- sc_placebo_test(task)

# p-value
task$results$placebo$p_value

# RMSPE ratios for all donor units
task$results$placebo$rmspe_ratios

You can limit the number of placebo units for faster computation:

task <- sc_placebo_test(task, n_placebo = 5)

References