stagsynth: Staggered Synthetic Control Estimation and Inference

Description

Implements the Staggered Synthetic Control (SSC) method of Cao, Lu, and Wu (2020) for estimating treatment effects in panel data with staggered adoption.

Main function

ssc: Estimate event-time ATT, overall ATT, and placebo-in-time confidence intervals.

Utilities

• panel_to_matrices: Convert long-format panel data to the N \times T matrices expected by ssc().

• ssc_min_eigenvalue: Check the design matrix invertibility condition.

• synthetic_control: Estimate SC weights for a single treated unit.

• synthetic_control_batch: Estimate SC weights for all units.

Author(s)

Maintainer: Zhanchao Fu fu.zhan@northeastern.edu

Authors:

• Jianfei Cao jianfeicao7@gmail.com

Convert Long-Format Panel Data to Matrices

Description

Transform a data frame in long format (one row per unit-period) into the N \times T matrices Y and D expected by ssc.

Usage

panel_to_matrices(data, unit, time, outcome, treatment)

Arguments

Value

A list with components

Y

Numeric N \times T outcome matrix.

D

Numeric N \times T treatment matrix.

units

Sorted vector of unique unit identifiers.

times

Sorted vector of unique time periods.

Examples

df <- data.frame(
  id   = rep(1:4, each = 6),
  time = rep(1:6, times = 4),
  Y    = rnorm(24),
  D    = c(rep(0, 12), rep(c(0,0,0,1,1,1), 2))
)
mat <- panel_to_matrices(df, unit = "id", time = "time",
                         outcome = "Y", treatment = "D")

Plot Event-Time ATT from SSC Estimation

Description

Plot Event-Time ATT from SSC Estimation

Usage

## S3 method for class 'ssc'
plot(
  x,
  main = "Event-time ATT (SSC)",
  xlab = "Event time",
  ylab = "ATT estimate",
  ci = !anyNA(x$ci_lower_event),
  ...
)

Arguments

Value

A ggplot object (invisibly) if ggplot2 is available; otherwise a base-R plot is drawn and NULL is returned invisibly.

Examples

set.seed(1)
N <- 10; Ttot <- 8
Y <- matrix(rnorm(N * Ttot), N, Ttot)
D <- matrix(0L, N, Ttot)
D[1:3, 5:Ttot] <- 1L   # units 1-3 treated from period 5
fit <- ssc(Y, D, S = 2, alpha = 0.05)
plot(fit)

Print an ssc Object

Description

Compact one-line summary of a "ssc" estimation result.

Usage

## S3 method for class 'ssc'
print(x, ...)

Arguments

Value

x, invisibly.

Staggered Synthetic Control Estimation

Description

Estimate treatment effects in a panel with staggered adoption using the Staggered Synthetic Control (SSC) method of Cao, Lu, and Wu (2020). Returns event-time ATT, overall ATT, heterogeneous treatment effects, and placebo-in-time confidence intervals.

Usage

ssc(Y, D, S = NULL, alpha = 0.05)

Arguments

Details

The SSC method proceeds in four steps:

1. SC weights. For every unit, estimate synthetic control weights from pre-treatment data.

2. Treatment structure. Build the treatment assignment matrices A_s that map heterogeneous effects \gamma to unit-level outcomes at each post-treatment period.

3. Estimation. Solve a GLS-type system to recover \hat\gamma, then aggregate to event-time or overall ATT via a linear map L.

4. Inference. Construct a null distribution by applying the same estimator to rolling windows of pre-treatment residuals (placebo-in-time). Confidence intervals are the \alpha/2 and 1 - \alpha/2 quantiles of this distribution, shifted by the point estimate.

Value

An object of class "ssc", a list containing:

att_event

Numeric vector of length S: event-time ATT estimates (averaged across units at each event time).

ci_lower_event, ci_upper_event

Numeric vectors of length S: lower and upper bounds of (1 - \alpha) placebo-in-time confidence intervals. NA when T < S (too few pre-treatment periods).

att_overall

Scalar: overall ATT (simple average of all heterogeneous effects).

ci_lower_overall, ci_upper_overall

Scalar CI bounds for the overall ATT. NA when T < S.

p_value

Two-sided p-value for H_0: ATT = 0 based on the placebo distribution. NA when T < S.

gamma_hat

Numeric vector of length K: heterogeneous treatment effects for every treated (unit, post-period) pair.

te_mat_hat

Numeric N \times S matrix of unit-level treatment effects at each post-treatment period.

B_hat

Numeric N \times N SC weight matrix.

a_hat

Numeric vector of length N: SC intercepts.

u_hat

Numeric N \times T matrix of pre-treatment SC residuals.

min_eigenvalue

Smallest eigenvalue of the sample analogue of the design matrix \sum_s A_s' \hat M A_s. Must be positive for the estimator to be well-defined.

index_mat

Integer K \times 3 matrix. Each row (s, i, e) records the post-treatment period s, unit i, and event time e for one element of \hat\gamma.

N, T, S, K

Panel dimensions.

alpha

Significance level used.

References

Cao, J., Lu, C., and Wu, Y. (2020). "Synthetic Control Inference for Staggered Adoption."

Examples

set.seed(1)
N <- 5; Ttot <- 15
Y <- matrix(rnorm(N * Ttot), N, Ttot)
D <- matrix(0L, N, Ttot)
D[1, 8:15] <- 1L
D[2, 10:15] <- 1L
result <- ssc(Y, D)
print(result)
summary(result)

Compute Smallest Eigenvalue of the SSC Design Matrix

Description

A diagnostic function that builds the SSC design matrix \sum_s A_s' \hat M A_s and returns its smallest eigenvalue. This matrix must be positive definite for SSC estimates to exist.

Usage

ssc_min_eigenvalue(Y, D, S = NULL)

Arguments

Details

A positive value means the SSC estimator is well-defined; a value near zero warns that identification is weak.

Value

A scalar: the smallest eigenvalue.

Examples

set.seed(1)
N <- 10; Ttot <- 8
Y <- matrix(rnorm(N * Ttot), N, Ttot)
D <- matrix(0L, N, Ttot)
D[1:3, 5:Ttot] <- 1L   # units 1-3 treated from period 5
ssc_min_eigenvalue(Y, D, S = 2)

Summarise an ssc Object

Description

Prints a detailed summary of a "ssc" estimation result, including design diagnostics, the overall ATT with confidence interval and p-value, and a table of event-time ATT estimates.

Usage

## S3 method for class 'ssc'
summary(object, ...)

Arguments

Value

object, invisibly.

Synthetic Control Weights for a Single Treated Unit

Description

Estimate synthetic control weights by solving a constrained quadratic program on demeaned pre-treatment outcomes: minimise \|\tilde Y_1 - \tilde X b\|^2 subject to \sum b_j = 1, \; b_j \ge 0, where \tilde Y_1 and \tilde X are time-demeaned series for the treated unit and controls.

Usage

synthetic_control(Y)

Arguments

Details

The QP is solved by solve.QP. A small ridge term (10^{-6} I) is added to the Hessian for numerical stability when T is close to or smaller than N-1.

Value

A list with components

a_hat

Scalar intercept \hat a = \bar Y_1 - \bar X' \hat b.

b_hat

Numeric vector of length N. Entry 1 is 0 (the treated unit's self-weight); entries 2, \dots, N are the non-negative weights summing to 1.

Examples

set.seed(1)
Y <- matrix(rnorm(5 * 20), 5, 20)  # 5 units, 20 pre-treatment periods
res <- synthetic_control(Y)
res$b_hat   # SC weights for unit 1

Synthetic Control Weights for All Units (Batch)

Description

For each unit in turn, treat that unit as the "treated" unit and estimate SC weights from the remaining units. This produces an N \times N weight matrix \hat B with zeros on the diagonal.

Usage

synthetic_control_batch(Y)

Arguments

Value

A list with components

a_hat

Numeric vector of length N: unit-level intercepts.

B_hat

Numeric N \times N matrix of SC weights. Row i contains the weights used to construct the synthetic control for unit i; B_{ii} = 0.

Examples

set.seed(1)
Y <- matrix(rnorm(5 * 20), 5, 20)
res <- synthetic_control_batch(Y)
res$B_hat   # N x N weight matrix