Experimental Synthetic Control Design — scm

Selects which units to assign to the treatment arm (and which to the control arm) in a planned experiment, following Abadie and Zhao (2026). Both sets of units are chosen by minimising the distance between their weighted-average predictor vectors and the population-average predictor vector \(\bar{X}\), so the resulting estimates are less susceptible to post-randomisation bias than pure random assignment.

Usage

scm_design(
  data,
  outcome,
  unit,
  time,
  T0,
  T_fit = NULL,
  m_min = 1L,
  m_max = 1L,
  f = NULL,
  predictors = NULL,
  design = c("base", "weakly_targeted", "unit_level"),
  beta = 1,
  xi = 1,
  alpha = 0.05,
  normalize = TRUE,
  max_subsets = 100000L
)

Arguments

data: Long-format data frame (one row per unit–time).
outcome: Name of the outcome column.
unit: Name of the unit identifier column.
time: Name of the time identifier column.
T0: Last pre-experimental period (a value present in the time column). Periods after T0 are the experimental periods.
T_fit: Number of fitting periods, counted from the start of the pre-experimental phase. Defaults to NULL, which uses all pre-experimental periods for fitting (no blank periods; inference disabled). When T_fit is smaller than the total number of pre-experimental periods, the remaining periods become blank periods used for inference.
m_min: Minimum number of units assigned to treatment (default 1).
m_max: Maximum number of units assigned to treatment (default 1).
f: Named numeric vector of population weights \(f_j\). Defaults to uniform weights \(1/J\). Will be normalised to sum to 1.
predictors: A list() of pred() specifications that define the predictor matrix \(X_j\). Defaults to NULL, which uses all fitting-period outcome values as predictors.
design: Design formulation: "base" (default), "weakly_targeted", or "unit_level".
beta: Trade-off parameter \(\beta > 0\) for the Weakly targeted design (default 1).
xi: Trade-off parameter \(\xi > 0\) for the Unit-level design (default 1).
alpha: Significance level for confidence intervals (default 0.05).
normalize: If TRUE (default), each row of the predictor matrix is divided by its cross-unit standard deviation before optimisation, so predictors measured on different scales contribute equally.
max_subsets: Maximum number of treatment-set candidates to evaluate before switching to random sampling (default 100 000).

Value

An object of class "scm_design" with components:

treated_units: unit identifiers selected for treatment
control_units: unit identifiers in the control pool
w: J-length weight vector for the synthetic treated unit (sums to 1)
v: J-length weight vector for the synthetic control unit (sums to 1)
tau_hat: estimated treatment effects for each experimental period
p_value: permutation p-value (NA when blank periods are unavailable)
ci_lower, ci_upper: per-period split-conformal confidence interval
Y_synth_tr, Y_synth_co: synthetic treated/control series (all periods)
estimate: ATT (mean of tau_hat)

Details

Three design formulations are available:

"base" (eq. 7): both the synthetic treated and the synthetic control independently target the population average \(\bar{X}\).
"weakly_targeted" (eq. 9): the synthetic treated targets \(\bar{X}\); the synthetic control targets the synthetic treated predictor vector (controlled by beta).
"unit_level" (eq. 10): each treated unit gets its own synthetic control; the aggregate control weight is a convex combination (controlled by xi).

Inference uses "blank periods" — pre-experimental periods whose outcomes were not used to estimate the weights. Set T_fit strictly smaller than the number of pre-experimental periods to enable the permutation test and split- conformal confidence intervals from Section 3 of Abadie and Zhao (2026).

References

Abadie, A. and Zhao, J. (2026). "Synthetic Controls for Experimental Design." MIT Working Paper.