sccic: Synthetic Control Changes-in-Changes Estimator

Description

Implements the Changes-in-Changes (CIC) estimator of Athey and Imbens (2006) doi:10.1111/j.1468-0262.2006.00668.x combined with synthetic control methods. Provides nonparametric estimation of the entire counterfactual distribution of outcomes for a treated group, allowing evaluation of average, quantile, and distributional treatment effects. Synthetic control weights are constructed via elastic net regularization to handle settings with many potential control units.

Implements the Changes-in-Changes (CIC) estimator of Athey and Imbens (2006) combined with synthetic control methods for causal inference in difference-in-differences settings.

Main functions

cic

Standard CIC estimator for two groups and two periods, with analytic and bootstrap inference.

sc_cic

Combined synthetic control + CIC estimator for settings with a single treated unit and multiple donor units.

Author(s)

Maintainer: Neil Hwang neil.hwang@bcc.cuny.edu

See Also

Useful links:

• https://github.com/neilhwang/sccic

• Report bugs at https://github.com/neilhwang/sccic/issues

Fit synthetic control via elastic net (internal)

Description

Fit synthetic control via elastic net (internal)

Usage

.fit_sc(y_treated, y_donors, pre_idx, post_idx, alpha, seed)

Run one cross-sectional CIC simulation (DGP: nonlinear)

Description

Run one cross-sectional CIC simulation (DGP: nonlinear)

Usage

.run_cs_sim(N_cell, N_cell_post, tau_true, dgp)

Run one SC-CIC simulation

Description

Run one SC-CIC simulation

Usage

.run_sc_sim(T_pre, T_post, J, tau_true, dgp, alpha, boot_iters)

Validate CIC Input Data

Description

Checks for common data issues and issues informative warnings. Called internally by cic when input is potentially problematic. Also available for manual use.

Usage

check_data(y_00, y_01, y_10, y_11)

Arguments

Value

Invisible TRUE. Produces warnings for potential issues.

Check Support Condition

Description

Warns if the treated unit's pre-treatment outcomes fall outside the range of the synthetic control's pre-treatment outcomes, which can cause the CIC distributional transport to extrapolate.

Usage

check_support(x)

Arguments

Value

Logical. TRUE if support condition is satisfied.

Changes-in-Changes Estimator

Description

Implements the Changes-in-Changes (CIC) estimator of Athey and Imbens (2006) for the average treatment effect on the treated in a two-group, two-period difference-in-differences setting.

Usage

cic(
  y_00,
  y_01,
  y_10,
  y_11,
  se = TRUE,
  boot = FALSE,
  boot_iters = 500L,
  seed = NULL
)

Arguments

Details

The CIC estimator constructs a counterfactual distribution for the treated group in the post-treatment period by applying the transformation:

Y^{N,CIC}_{11} = F^{-1}_{Y,01}(F_{Y,00}(Y_{10}))

The average treatment effect is then:

\hat{\tau}^{CIC} = \frac{1}{N_{11}} \sum Y_{11,i} - \frac{1}{N_{10}} \sum F^{-1}_{Y,01}(F_{Y,00}(Y_{10,i}))

The analytic variance follows Theorem 5.1 of Athey and Imbens (2006):

Var(\sqrt{N} \hat{\tau}^{CIC}) = V^p/\alpha_{00} + V^q/\alpha_{01} + V^r/\alpha_{10} + V^s/\alpha_{11}

Value

An object of class "cic" containing:

References

Athey, S. and Imbens, G. W. (2006). Identification and Inference in Nonlinear Difference-in-Differences Models. Econometrica, 74(2), 431–497. doi:10.1111/j.1468-0262.2006.00668.x

Examples

# Workers' compensation example (Meyer, Viscusi, and Durbin 1995)
if (requireNamespace("wooldridge", quietly = TRUE)) {
  data("injury", package = "wooldridge")
  result <- cic(
    y_00 = injury$ldurat[injury$highearn == 0 & injury$afchnge == 0],
    y_01 = injury$ldurat[injury$highearn == 0 & injury$afchnge == 1],
    y_10 = injury$ldurat[injury$highearn == 1 & injury$afchnge == 0],
    y_11 = injury$ldurat[injury$highearn == 1 & injury$afchnge == 1]
  )
  print(result)
}

Compute Variance Components for CIC Inference

Description

Implements the analytic asymptotic variance from Theorem 5.1 of Athey and Imbens (2006). The variance has four components corresponding to the four group-period subsamples.

Usage

compute_variance_components(y_00, y_01, y_10, y_11, ecdfs)

Arguments

Value

A list with components V_p, V_q, V_r, V_s.

Estimate density at a point using finite differences

Description

Uses the approach from Athey and Imbens (2006), Section 5.

Usage

ecdf_density(ec, y, bandwidth = NULL)

Arguments

Value

Estimated density f(y).

Evaluate empirical CDF at a point

Description

Evaluate empirical CDF at a point

Usage

ecdf_eval(ec, y)

Arguments

Value

The empirical CDF value F(y).

Evaluate inverse empirical CDF (quantile function)

Description

Computes F^{-1}(q) = \inf\{y : F(y) \ge q\}.

Usage

ecdf_inv(ec, q)

Arguments

Value

The smallest value y such that F(y) >= q.

Empirical CDF and Inverse CDF Utilities

Description

Internal functions for computing empirical distribution functions and their inverses, used by the CIC estimator.

Leave-One-Out Donor Analysis

Description

Re-estimates SC-CIC dropping one donor at a time to assess sensitivity to individual donors.

Usage

loo_donors(y_treated, y_donors, treatment_period, alpha = 1, seed = 42)

Arguments

Value

A data frame with one row per donor, showing the SC-CIC estimate when that donor is excluded.

Compute empirical CDF from a sample

Description

Returns sorted unique values and their empirical CDF values.

Usage

make_ecdf(x)

Arguments

Value

A list with components:

Plot Pre-treatment Fit for SC-CIC

Description

Plots the treated unit against the synthetic control over time, with a vertical line at the treatment period.

Usage

## S3 method for class 'sc_cic'
plot(x, ...)

Arguments

Details

The plot shows the treated unit (solid line) and synthetic control (dashed line) over all time periods, with a vertical dashed line marking the start of treatment. Good pre-treatment fit is a necessary (but not sufficient) condition for valid SC-CIC inference.

Value

Invisible. Called for its side effect of producing a plot.

Plot Counterfactual Distribution

Description

Plots the empirical CDFs of the four group-period cells used in the CIC estimator, illustrating the distributional transport.

Usage

plot_distributions(x, ...)

Arguments

Value

Invisible. Called for its side effect of producing a plot.

Q-Q Plot of Treated vs Synthetic Control (Pre-treatment)

Description

Produces a quantile-quantile plot comparing the pre-treatment distributions of the treated unit and the synthetic control. This assesses whether the synthetic control tracks the treated unit's distributional dynamics, not just its mean—a necessary condition for CIC validity. Points on the 45-degree line indicate identical distributions.

Usage

plot_qq(x, ...)

Arguments

Value

Invisible. Called for its side effect of producing a plot.

Plot Quantile Treatment Effects

Description

Plot Quantile Treatment Effects

Usage

plot_qte(x, probs = seq(0.05, 0.95, 0.05), ...)

Arguments

Value

Invisible. Called for its side effect of producing a plot.

Compute Quantile Treatment Effects

Description

Estimates quantile treatment effects from a CIC fit by comparing quantiles of the actual post-treatment treated distribution with quantiles of the counterfactual distribution.

Usage

quantile_te(x, probs = seq(0.05, 0.95, 0.05))

Arguments

Details

The quantile treatment effect at quantile q is:

\hat{\tau}_q = \hat{F}^{-1}_{Y^I,11}(q) - \hat{F}^{-1}_{Y^N,11}(q)

where \hat{F}^{-1}_{Y^N,11} is the CIC counterfactual distribution.

Value

A data frame with columns quantile, actual, counterfactual, and qte (quantile treatment effect).

Synthetic Control Changes-in-Changes Estimator

Description

Combines synthetic control methods with the Changes-in-Changes estimator. First constructs a synthetic control unit from donor units using elastic net regularization, then applies the CIC estimator using the synthetic control as the comparison group.

Usage

sc_cic(
  y_treated,
  y_donors,
  treatment_period,
  alpha = 1,
  boot = TRUE,
  boot_iters = 500L,
  seed = NULL
)

Arguments

Details

The procedure works in two steps:

Step 1: Synthetic Control Construction. In the pre-treatment period, the treated unit's outcome is regressed on the donor units' outcomes using elastic net (via cv.glmnet). This yields a sparse set of weights that construct a synthetic control unit as a weighted combination of donors.

Step 2: CIC Estimation. The CIC estimator is applied with the synthetic control as the "control group" and the treated unit as the "treatment group."

Inference. Because the synthetic control is an estimated object, the analytic asymptotic variance of Athey and Imbens (2006) does not directly apply. Instead, sc_cic provides bootstrap standard errors that re-estimate the elastic net weights in each bootstrap iteration, thereby accounting for first-stage estimation uncertainty. The bootstrap resamples time periods (with replacement) within the pre-treatment and post-treatment windows separately, preserving the panel structure.

Value

An object of class "sc_cic" inheriting from "cic", with components:

References

Athey, S. and Imbens, G. W. (2006). Identification and Inference in Nonlinear Difference-in-Differences Models. Econometrica, 74(2), 431–497.

Abadie, A., Diamond, A., and Hainmueller, J. (2010). Synthetic Control Methods for Comparative Case Studies. Journal of the American Statistical Association, 105(490), 493–505.

Examples

# Basque Country example
if (requireNamespace("Synth", quietly = TRUE)) {
  data("basque", package = "Synth")
  gdp <- reshape(basque[, c("regionno", "year", "gdpcap")],
                 idvar = "year", timevar = "regionno", direction = "wide")
  y_treated <- gdp[, "gdpcap.17"]
  donors <- as.matrix(gdp[, grep("gdpcap\\.", names(gdp))])
  donors <- donors[, !colnames(donors) %in% c("gdpcap.17", "gdpcap.1")]
  valid <- complete.cases(y_treated, donors)
  result <- sc_cic(y_treated[valid], donors[valid, ],
                   treatment_period = 16, seed = 42)
  print(result)
}

Extract Synthetic Control Weights

Description

Returns a data frame of donor weights from an SC-CIC fit, sorted by absolute weight. Useful for inspecting which donors contribute to the synthetic control.

Usage

sc_weights(x, nonzero_only = TRUE)

Arguments

Value

A data frame with columns donor and weight.

Sensitivity Analysis for SC-CIC

Description

Re-estimates the SC-CIC treatment effect over a grid of elastic net penalty parameters, showing sensitivity to the regularization choice.

Usage

sensitivity_alpha(
  y_treated,
  y_donors,
  treatment_period,
  alphas = seq(0, 1, 0.2),
  seed = 42
)

Arguments

Value

A data frame with columns alpha, tau_cic, tau_did, n_donors, and pre_rmse.

Simulation Study for SC-CIC

Description

Generates data under controlled DGPs and evaluates SC-CIC performance.

Usage

simulate_sccic(
  n_sims = 500,
  T_pre = 25,
  T_post = 15,
  J = 15,
  tau_true = 1,
  dgp = c("linear", "nonlinear", "sc_good", "sc_bad"),
  alpha = 1,
  boot_iters = 200,
  seed = 42,
  verbose = TRUE
)

Arguments

Details

Four DGPs are available, designed to test different aspects of SC-CIC:

DGP 1: "linear" — Baseline. Outcomes are linear in a common factor and unit-specific loadings. DID is correctly specified. CIC matches DID. SC fits well. Purpose: verify the method works in the easy case.

DGP 2: "nonlinear" — CIC advantage. Cross-sectional DGP (not SC). N observations per cell. Control and treated have different distributions of unobservables. The production function is nonlinear and changes over time. DID is biased due to the nonlinear distributional shift; CIC is correct. Purpose: demonstrate the advantage of CIC over DID. Note: this tests cic(), not sc_cic().

DGP 3: "sc_good" — SC with good distributional fit. The treated unit is a true sparse combination of donors plus noise. SC recovers the weights well; the distributional dynamics are similar. Purpose: show SC-CIC works when SC fit is good.

DGP 4: "sc_bad" — SC with mean-only fit. The SC matches the treated mean, but donors have much lower variance than the treated unit. The distributional transport is wrong. Purpose: show SC-CIC fails when distributional assumptions are violated.

Value

A data frame with simulation results.

Examples

# Quick example (runs in seconds)
r <- simulate_sccic(n_sims = 2, dgp = "nonlinear", tau_true = 1, boot_iters = 5, verbose = FALSE)
summarize_simulation(r, tau_true = 1)


# Full simulation
r <- simulate_sccic(n_sims = 200, dgp = "nonlinear", tau_true = 1)
summarize_simulation(r, tau_true = 1)

Summarize simulation results

Description

Summarize simulation results

Usage

summarize_simulation(results, tau_true)

Arguments

Value

Prints summary statistics and returns them invisibly.