Synthetic Control Methods for Strategic Policy Interventions

causal-inference
synthetic-control
policy-evaluation
Apply the synthetic control method to estimate the causal effect of a competition policy change in a strategic environment, using constrained optimisation to construct counterfactual outcomes from untreated regions.
Author

Raban Heller

Published

May 8, 2026

Modified

May 8, 2026

Keywords

synthetic control, causal inference, competition policy, counterfactual, panel data

Introduction & motivation

One of the most persistent challenges in policy evaluation is the fundamental problem of causal inference: we can never observe what would have happened to a treated unit had it not been treated. This problem becomes even more acute in strategic environments, where economic agents – firms, regulators, countries – respond to policy changes by adjusting their behaviour in anticipation of competitors doing the same. When a government reforms its competition policy, the effects ripple through markets in ways that depend critically on the strategic interactions among firms. Traditional difference-in-differences methods rely on parallel trends assumptions that are difficult to justify in such settings, because strategic responses can cause treated and control units to diverge in the pre-treatment period in subtle, hard-to-detect ways.

The synthetic control method (SCM), developed by Abadie and Gardeazabal (2003) and formalised by Abadie, Diamond, and Hainmueller (2010), offers an elegant solution. Instead of relying on a single control unit or the simple average of several controls, SCM constructs a weighted combination of untreated units – a “synthetic control” – that best reproduces the pre-treatment trajectory of the treated unit. The weights are chosen through constrained optimisation: they must be non-negative and sum to one, which ensures that the synthetic control is a convex combination of real units rather than an extrapolation beyond the support of the data. This data-driven approach to constructing counterfactuals has been called “arguably the most important innovation in the policy evaluation literature in the last 15 years” by Athey and Imbens (2017).

In the context of game theory and strategic interactions, SCM is particularly valuable because it allows us to evaluate policy interventions that alter the rules of the game. Consider a region that introduces a stricter merger control regime, effectively changing the payoff structure for firms contemplating horizontal mergers. The direct effect on market concentration is straightforward to hypothesise, but the equilibrium effects – how firms adjust their investment, pricing, and entry decisions in response – are far more complex. By constructing a synthetic version of the treated region from regions that did not change their merger policy, we can estimate the total causal effect, including all equilibrium adjustments.

The logic behind SCM connects deeply to ideas in game theory. The method essentially solves an optimisation problem that is analogous to finding a mixed strategy: just as a mixed Nash equilibrium assigns probabilities to pure strategies to make the opponent indifferent, SCM assigns weights to control units to best match the treated unit’s pre-treatment outcomes. The constraint that weights be non-negative and sum to one mirrors the requirement that a probability distribution lies on the simplex. Moreover, the placebo tests central to SCM inference – where the method is applied to each control unit in turn, pretending it was treated – are conceptually similar to checking whether a proposed equilibrium is robust to deviations by individual players.

This tutorial implements the synthetic control method from scratch in R, applying it to simulated panel data from a strategic environment. We generate data where multiple regions host firms that compete in a Cournot oligopoly, and one region introduces a competition policy reform that changes the cost structure. We then use SCM to estimate the causal effect of this reform on total market output, demonstrating the full pipeline: data simulation, weight estimation via constrained optimisation, gap analysis, and placebo inference. By building SCM from first principles using only base R optimisation tools, we gain insight into both the mechanics of the method and its strategic foundations.

Mathematical formulation

Consider \(J + 1\) regions observed over \(T\) time periods, where region \(j = 1\) receives the treatment at time \(T_0 + 1\). Regions \(j = 2, \ldots, J+1\) form the donor pool of potential controls. Let \(Y_{jt}\) denote the outcome of interest for region \(j\) at time \(t\).

The synthetic control estimator seeks a vector of weights \(\mathbf{w} = (w_2, \ldots, w_{J+1})'\) that solves:

\[ \min_{\mathbf{w}} \sum_{t=1}^{T_0} \left( Y_{1t} - \sum_{j=2}^{J+1} w_j Y_{jt} \right)^2 \]

subject to the constraints:

\[ w_j \geq 0 \quad \forall \, j \in \{2, \ldots, J+1\}, \qquad \sum_{j=2}^{J+1} w_j = 1. \]

The estimated treatment effect at post-treatment time \(t > T_0\) is:

\[ \hat{\tau}_t = Y_{1t} - \sum_{j=2}^{J+1} w_j^* Y_{jt} = Y_{1t} - Y_{1t}^{\text{synth}} \]

where \(w_j^*\) are the optimal weights and \(Y_{1t}^{\text{synth}}\) is the synthetic control prediction.

For placebo inference, we apply the same procedure to each donor unit \(k \in \{2, \ldots, J+1\}\), treating it as if it were the treated unit. Define the pre-treatment root mean squared prediction error (RMSPE) and post-treatment RMSPE:

\[ \text{RMSPE}_{\text{pre}}^{(k)} = \sqrt{\frac{1}{T_0} \sum_{t=1}^{T_0} \hat{\tau}_{t}^{(k)2}}, \qquad \text{RMSPE}_{\text{post}}^{(k)} = \sqrt{\frac{1}{T - T_0} \sum_{t=T_0+1}^{T} \hat{\tau}_{t}^{(k)2}} \]

The ratio \(\text{RMSPE}_{\text{post}}^{(k)} / \text{RMSPE}_{\text{pre}}^{(k)}\) provides a test statistic. The p-value is the fraction of units (including the treated unit) whose ratio is at least as large as that of the treated unit.

In our strategic setting, the outcome \(Y_{jt}\) is total market output in region \(j\), generated by a Cournot oligopoly with \(n\) firms. Each firm \(i\) in region \(j\) at time \(t\) chooses output \(q_{ijt}\) to maximise profit \(\pi_{ijt} = (a - Q_{jt}) q_{ijt} - c_{jt} q_{ijt}\), where \(Q_{jt} = \sum_i q_{ijt}\). The Nash equilibrium output for each firm is \(q^* = (a - c_{jt}) / (n + 1)\), yielding total output \(Y_{jt} = n(a - c_{jt})/(n+1) + \varepsilon_{jt}\).

R implementation

We simulate panel data for 10 regions over 30 periods. Region 1 receives a policy intervention at period 16 that reduces production costs, leading to higher equilibrium output. We then estimate synthetic control weights using optim() with the L-BFGS-B method, which supports box constraints, combined with a projection step to enforce the simplex constraint.

set.seed(42)

# --- Parameters ---
J_plus_1 <- 10        # total regions
T_total  <- 30        # total periods
T0       <- 15        # treatment occurs after period 15
n_firms  <- 4         # firms per region (Cournot)
a        <- 100       # demand intercept
treat_effect <- -10   # cost reduction in treated region post-treatment

# --- Generate region-specific cost levels (persistent heterogeneity) ---
base_costs <- round(runif(J_plus_1, 30, 60), 1)

# --- Simulate panel data ---
panel <- expand.grid(region = 1:J_plus_1, period = 1:T_total) |>
  mutate(
    base_cost = base_costs[region],
    cost_shock = rnorm(n(), 0, 2),
    time_trend = 0.3 * period,
    cost = base_cost + cost_shock + time_trend,
    # Treatment: region 1 gets cost reduction after T0
    treatment = as.integer(region == 1 & period > T0),
    cost = cost + treatment * treat_effect,
    # Cournot equilibrium total output
    eq_output = n_firms * (a - cost) / (n_firms + 1),
    # Add measurement noise
    output = eq_output + rnorm(n(), 0, 1.5)
  )

cat("Panel dimensions:", J_plus_1, "regions x", T_total, "periods\n")
Panel dimensions: 10 regions x 30 periods
cat("Treatment region: 1, treatment period:", T0 + 1, "\n")
Treatment region: 1, treatment period: 16 
cat("Base costs by region:\n")
Base costs by region:
cat(paste(paste0("  Region ", 1:J_plus_1, ": ", base_costs), collapse = "\n"), "\n")
  Region 1: 57.4
  Region 2: 58.1
  Region 3: 38.6
  Region 4: 54.9
  Region 5: 49.3
  Region 6: 45.6
  Region 7: 52.1
  Region 8: 34
  Region 9: 49.7
  Region 10: 51.2 
# --- Prepare matrices ---
Y_mat <- panel |>
  select(region, period, output) |>
  pivot_wider(names_from = region, values_from = output) |>
  select(-period) |>
  as.matrix()

Y_treated <- Y_mat[, 1]
Y_donors  <- Y_mat[, -1]

# --- SCM weight estimation via constrained optimisation ---
# Project weights onto the simplex after each evaluation
project_simplex <- function(v) {
  v <- pmax(v, 0)
  s <- sum(v)
  if (s == 0) return(rep(1 / length(v), length(v)))
  v / s
}

scm_objective <- function(w_raw) {
  w <- project_simplex(w_raw)
  Y_synth_pre <- Y_donors[1:T0, ] %*% w
  sum((Y_treated[1:T0] - Y_synth_pre)^2)
}

# Optimise
n_donors <- ncol(Y_donors)
w_init <- rep(1 / n_donors, n_donors)
opt_result <- optim(
  par = w_init, fn = scm_objective,
  method = "L-BFGS-B",
  lower = rep(0, n_donors),
  upper = rep(1, n_donors),
  control = list(maxit = 5000)
)
w_star <- project_simplex(opt_result$par)

cat("\nOptimal SCM weights:\n")

Optimal SCM weights:
for (i in seq_along(w_star)) {
  if (w_star[i] > 0.001) {
    cat(sprintf("  Region %d: %.4f\n", i + 1, w_star[i]))
  }
}
  Region 2: 0.6358
  Region 4: 0.3642
# --- Compute synthetic control and treatment effects ---
Y_synth <- as.numeric(Y_donors %*% w_star)
gap <- Y_treated - Y_synth

cat("\nPre-treatment RMSPE:", round(sqrt(mean(gap[1:T0]^2)), 3), "\n")

Pre-treatment RMSPE: 2.486 
cat("Post-treatment avg gap:", round(mean(gap[(T0+1):T_total]), 3), "\n")
Post-treatment avg gap: 7.435 
# --- Placebo tests ---
placebo_gaps <- matrix(NA, nrow = T_total, ncol = n_donors)
placebo_ratios <- numeric(n_donors)

for (k in 1:n_donors) {
  Y_placebo_treated <- Y_donors[, k]
  Y_placebo_donors  <- cbind(Y_mat[, 1], Y_donors[, -k])

  scm_placebo <- function(w_raw) {
    w <- project_simplex(w_raw)
    Y_s <- Y_placebo_donors[1:T0, ] %*% w
    sum((Y_placebo_treated[1:T0] - Y_s)^2)
  }

  n_p <- ncol(Y_placebo_donors)
  opt_p <- optim(
    par = rep(1/n_p, n_p), fn = scm_placebo,
    method = "L-BFGS-B",
    lower = rep(0, n_p), upper = rep(1, n_p),
    control = list(maxit = 5000)
  )
  w_p <- project_simplex(opt_p$par)
  Y_synth_p <- as.numeric(Y_placebo_donors %*% w_p)
  placebo_gaps[, k] <- Y_placebo_treated - Y_synth_p

  rmspe_pre  <- sqrt(mean(placebo_gaps[1:T0, k]^2))
  rmspe_post <- sqrt(mean(placebo_gaps[(T0+1):T_total, k]^2))
  placebo_ratios[k] <- rmspe_post / max(rmspe_pre, 0.01)
}

# Treated unit ratio
rmspe_pre_treat  <- sqrt(mean(gap[1:T0]^2))
rmspe_post_treat <- sqrt(mean(gap[(T0+1):T_total]^2))
treat_ratio <- rmspe_post_treat / max(rmspe_pre_treat, 0.01)

p_value <- mean(c(treat_ratio, placebo_ratios) >= treat_ratio)
cat("RMSPE ratio (treated):", round(treat_ratio, 3), "\n")
RMSPE ratio (treated): 3.156 
cat("Placebo p-value:", round(p_value, 3), "\n")
Placebo p-value: 0.1

Static publication-ready figure

The gap plot below shows the difference between observed and synthetic control outcomes over time. The vertical dashed line marks the treatment period. In the pre-treatment period, the gap should hover near zero (indicating a good fit), while the post-treatment gap reveals the estimated causal effect.

# Build data frame for plotting
gap_df <- data.frame(
  period = rep(1:T_total, times = 1 + n_donors),
  gap = c(gap, as.numeric(placebo_gaps)),
  unit = rep(c("Treated (Region 1)", paste0("Placebo ", 1:n_donors)),
             each = T_total),
  is_treated = rep(c(TRUE, rep(FALSE, n_donors)), each = T_total)
)

p_static <- ggplot(gap_df, aes(x = period, y = gap, group = unit)) +
  geom_vline(xintercept = T0, linetype = "dashed", color = "grey50", linewidth = 0.5) +
  geom_hline(yintercept = 0, color = "grey70", linewidth = 0.3) +
  geom_line(data = filter(gap_df, !is_treated),
            color = okabe_ito[8], alpha = 0.4, linewidth = 0.4) +
  geom_line(data = filter(gap_df, is_treated),
            color = okabe_ito[1], linewidth = 1.2) +
  annotate("text", x = T0 - 0.5, y = max(gap_df$gap) * 0.9,
           label = "Treatment", hjust = 1, size = 3.5, color = "grey40") +
  labs(
    title = "Synthetic Control Gap Plot: Policy Intervention Effect",
    subtitle = "Treated region vs. synthetic control (orange) with placebo gaps (grey)",
    x = "Period", y = "Gap (Observed - Synthetic)"
  ) +
  theme_publication()

p_static
Figure 1: Figure 1. Synthetic control gap plot for strategic policy intervention. The vertical dashed line at period 15 marks the introduction of the competition policy reform. The treated region’s output diverges upward from its synthetic control post-treatment, consistent with the cost-reducing effect of the policy. Placebo gaps (grey) from donor regions show no comparable divergence. Simulated Cournot oligopoly data.

Interactive figure

The interactive version allows hovering over each line to identify the unit and inspect gap values at each time period.

gap_df <- gap_df |>
  mutate(
    tooltip_text = paste0(
      "Unit: ", unit, "\n",
      "Period: ", period, "\n",
      "Gap: ", round(gap, 2)
    )
  )

p_interactive <- ggplot(gap_df, aes(x = period, y = gap, group = unit, text = tooltip_text)) +
  geom_vline(xintercept = T0, linetype = "dashed", color = "grey50", linewidth = 0.5) +
  geom_hline(yintercept = 0, color = "grey70", linewidth = 0.3) +
  geom_line(data = filter(gap_df, !is_treated),
            color = okabe_ito[8], alpha = 0.4, linewidth = 0.4) +
  geom_line(data = filter(gap_df, is_treated),
            color = okabe_ito[1], linewidth = 1.2) +
  labs(
    title = "Synthetic Control Gap Plot (Interactive)",
    x = "Period", y = "Gap (Observed - Synthetic)"
  ) +
  theme_publication()

ggplotly(p_interactive, tooltip = "text") |>
  config(displaylogo = FALSE, modeBarButtonsToRemove = c("select2d", "lasso2d"))
Figure 2

Interpretation

The synthetic control analysis reveals a clear and statistically significant effect of the competition policy reform on market output in the treated region. After the intervention at period 16, observed output in Region 1 diverges sharply upward from its synthetic counterpart, with an average post-treatment gap that reflects the equilibrium response of Cournot competitors to lower production costs. This gap is not merely a statistical artefact: the placebo tests demonstrate that no donor region exhibits a comparable divergence, yielding a small p-value that allows us to reject the null hypothesis of no treatment effect.

The mechanics of the result connect directly to game-theoretic reasoning. In a Cournot oligopoly with \(n\) symmetric firms and linear demand, total equilibrium output is \(Q^* = n(a-c)/(n+1)\). A reduction in marginal cost \(c\) by \(\Delta c\) increases total output by \(n \cdot |\Delta c| / (n+1)\). With four firms and our parameterised cost reduction of 10 units, the predicted effect is \(4 \times 10 / 5 = 8\) units of output – a figure that our SCM estimate should approximate closely, provided the synthetic control adequately matches the pre-treatment trajectory. The close correspondence between the theoretical prediction and the estimated gap validates both the SCM methodology and the underlying game-theoretic model.

Several aspects of the implementation merit careful discussion. First, the weight estimation uses a simplex projection approach rather than a dedicated quadratic programming solver. While packages like Synth or augsynth provide more sophisticated implementations with predictor-matching and cross-validation, our from-scratch approach using optim() with L-BFGS-B illustrates the core optimisation problem transparently. The simplex constraint – weights non-negative and summing to one – is crucial because it prevents extrapolation: the synthetic control is always a weighted average of real observed units, never a fantasy projection beyond the data.

Second, the placebo inference framework provides a non-parametric approach to statistical testing that does not rely on large-sample asymptotics. In our setting with 10 regions, the smallest achievable p-value is 0.1 (when the treated unit has the most extreme RMSPE ratio). This limitation is inherent to the method and highlights the importance of having a sufficiently large donor pool. In real applications, researchers sometimes restrict the donor pool to units with good pre-treatment fit (small pre-treatment RMSPE), which can sharpen inference but also raises concerns about cherry-picking.

Third, the strategic context introduces subtleties that are absent from standard SCM applications. When a region changes its competition policy, firms in that region adjust their strategies, but so might firms in neighbouring regions – through spatial competition, supply chain linkages, or anticipatory regulatory arbitrage. Such interference between units violates the stable unit treatment value assumption (SUTVA) that underlies causal inference. In our simulation, we assumed no cross-regional spillovers, but in practice, researchers must carefully consider whether the treatment in one region affects outcomes in donor regions. If it does, the synthetic control will be contaminated, and the estimated effect will be biased.

The gap plot – the primary visual output of any SCM analysis – is both a diagnostic tool and a persuasion device. In the pre-treatment period, a small gap indicates that the synthetic control successfully reproduces the treated unit’s trajectory, lending credibility to the counterfactual projections in the post-treatment period. The simultaneous display of placebo gaps provides immediate visual evidence for or against the treatment effect: if the treated unit’s gap is clearly an outlier relative to the placebos, the effect is likely genuine. This visual inference approach has been influential in applied economics precisely because it communicates causal evidence in an intuitive, transparent way that does not hide behind opaque test statistics.

Finally, it is worth noting the connections between SCM and other approaches to causal inference in strategic settings. Difference-in-differences requires parallel trends and benefits from having many treated and control units. Regression discontinuity exploits threshold rules. Instrumental variables require exogenous variation in the strategic environment. SCM occupies a unique niche: it is designed for situations where a single unit (or a small number of units) is treated, and the researcher has access to a panel of untreated units. This makes it particularly well-suited for evaluating national or regional policy changes, regulatory reforms, and other interventions that affect the rules of strategic interaction at a macro level.

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Citation

BibTeX citation:
@online{heller2026,
  author = {Heller, Raban},
  title = {Synthetic {Control} {Methods} for {Strategic} {Policy}
    {Interventions}},
  date = {2026-05-08},
  url = {https://r-heller.github.io/equilibria/tutorials/causal-inference/synthetic-control-strategic/},
  langid = {en}
}
For attribution, please cite this work as:
Heller, Raban. 2026. “Synthetic Control Methods for Strategic Policy Interventions.” May 8. https://r-heller.github.io/equilibria/tutorials/causal-inference/synthetic-control-strategic/.