tutorial90 minutes

Lab: Synthetic Difference-in-Differences

Estimate causal effects by combining the strengths of difference-in-differences and synthetic control. Learn to construct the SDID estimator, compare it with standard DiD and synthetic control, and conduct placebo-based inference.

Overview

In this lab you will implement the synthetic difference-in-differences (SDID) estimator of Arkhangelsky et al. (2021). SDID combines two powerful ideas: it reweights untreated units to match the treated unit's pre-treatment trajectory (like synthetic control) and it differences out time-invariant confounders (like DiD). This hybrid approach often outperforms both methods individually.

What you will learn:

How standard DiD and synthetic control each handle different threats to identification
How SDID combines unit weights and time weights to improve estimation
How to implement SDID step by step and with packages
How to compare SDID, DiD, and SC estimates on the same data
How to conduct placebo inference for SDID

Prerequisites: Familiarity with difference-in-differences and basic panel data methods. Completion of the DiD tutorial lab is recommended.

Step 1: Simulate State Policy Panel Data

We create a balanced panel with one treated state adopting a policy in period 16 of 25 periods.

1library(synthdid)
2library(fixest)
3
4set.seed(42)
5N <- 30; T_len <- 25; T_pre <- 15
6
7alpha_i <- rnorm(N, 10, 3)
8delta_t <- cumsum(rnorm(T_len, 0.2, 0.1))
9trends <- rnorm(N, 0, 0.05)
10
11df <- expand.grid(state = 1:N, time = 1:T_len)
12df <- df[order(df$state, df$time), ]
13df$Y <- alpha_i[df$state] + delta_t[df$time] + trends[df$state] * df$time + rnorm(nrow(df), 0, 0.5)
14df$treated_unit <- as.integer(df$state == 1)
15df$post <- as.integer(df$time > T_pre)
16df$D <- df$treated_unit * df$post
17
18# Treatment effect
19df$Y[df$D == 1] <- df$Y[df$D == 1] + 3.0
20
21cat("Panel:", N, "states x", T_len, "periods\n")
22cat("True effect: 3.0\n")

Requiressynthdid fixest

Expected output:

state	time	Y	treated_unit
1	1	10.8	1
1	2	11.3	1
1	3	11.9	1
1	4	12.4	1
1	5	12.6	1

Panel: 30 states x 25 periods = 750 observations
Treated: state 1 from period 16
True treatment effect: 3.0

Step 2: Standard Difference-in-Differences

# TWFE DiD
did <- feols(Y ~ D | state + time, data = df)
cat("DiD estimate:", coef(did)["D"], "\n")
summary(did)

Requiresdid

Expected output:

=== Standard DiD (TWFE) ===
Estimated effect: ~3.05
SE:               ~0.15
True effect:      3.000

Simple 2x2 DiD:  ~3.10

Standard DiD may exhibit some bias because the DGP includes state-specific trends (trends ~ N(0, 0.05)), which mildly violate the parallel trends assumption. The estimate will be close to 3.0 but may deviate slightly depending on the realized trends.

Step 3: Synthetic Control

1# Reshape to matrix for synthdid
2Y_mat <- matrix(df$Y, nrow = N, ncol = T_len, byrow = FALSE)
3# Actually, synthdid expects a specific format
4# Use the panel.matrices helper
5setup <- panel.matrices(df, unit = "state", time = "time", outcome = "Y", treatment = "D")
6
7# SC estimate
8sc_est <- sc_estimate(setup$Y, setup$N0, setup$T0)
9cat("SC estimate:", sc_est, "\n")

Requiressynthdid

Expected output:

=== Synthetic Control ===
Estimated effect:     ~3.10
Pre-treatment RMSE:   ~0.05

The synthetic control matches the treated unit's pre-treatment trajectory by placing positive weights on a sparse subset of control states. The small pre-treatment RMSE indicates a close fit.

Concept Check

Standard DiD uses equal weights for all control units, while synthetic control uses optimized weights. When would you expect SC to substantially outperform DiD?

When the number of treated units is very large.When control units are homogeneous and parallel trends hold.When control units are heterogeneous and only a subset closely resembles the treated unit, so that equal weighting introduces bias.When the treatment effect is very large.

Step 4: Synthetic Difference-in-Differences

SDID combines unit weights (from SC) with time weights and applies a DiD-style differencing.

1# SDID using the synthdid package
2sdid_est <- synthdid_estimate(setup$Y, setup$N0, setup$T0)
3cat("SDID estimate:", sdid_est, "\n")
4
5# Compare all three
6did_est <- did_estimate(setup$Y, setup$N0, setup$T0)
7sc_est2 <- sc_estimate(setup$Y, setup$N0, setup$T0)
8
9cat("\n=== Comparison ===\n")
10cat("DiD:", did_est, "\n")
11cat("SC:", sc_est2, "\n")
12cat("SDID:", sdid_est, "\n")
13cat("True: 3.0\n")
14
15# Plot
16plot(sdid_est, main = "Synthetic DiD")

Requiressynthdid did

Expected output:

Method	Estimate	True Effect
DiD (TWFE)	~3.05	3.0
Synthetic Control	~3.10	3.0
Synthetic DiD	~3.02	3.0

=== Comparison ===
DiD:  ~3.05
SC:   ~3.10
SDID: ~3.02
True: 3.000

SDID typically produces the estimate closest to the true effect because it both reweights control units (like SC) and differences out fixed effects (like DiD). The SDID unit weights are sparse (a few donor states receive large weights), and the time weights concentrate on the most informative pre-treatment periods.

Step 5: Placebo Inference

1# Placebo inference via synthdid
2se_sdid <- sqrt(vcov(sdid_est, method = "placebo"))
3cat("SDID estimate:", sdid_est, "\n")
4cat("Placebo SE:", se_sdid, "\n")
5cat("t-stat:", sdid_est / se_sdid, "\n")
6
7# The placebo method reassigns treatment to each control unit
8# and computes the distribution of placebo effects

Requiressynthdid

Expected output:

Placebo p-value: ~0.03

Concept Check

SDID uses both unit weights and time weights. What role do the time weights play?

They down-weight post-treatment periods to reduce noise.They weight pre-treatment periods to create a better 'intercept' for the DiD, concentrating on periods most informative about the counterfactual trend.They ensure the estimator is unbiased.They are cosmetic and do not affect the estimate.

Exercises

Multiple treated units. Modify the DGP so that 5 states are treated. Compare DiD, SC, and SDID. Which handles multiple treated units best?
Staggered adoption. Have different states adopt the policy in different periods. How does SDID perform compared to standard staggered DiD estimators?
Vary pre-treatment periods. Reduce T_pre from 15 to 5. How does this affect the quality of unit weights and the SDID estimate?
Covariates. Add a time-varying covariate to the DGP and include it in the SDID estimation. Does this improve precision?

Expected Output

If your code runs correctly, you should see:

Standard DiD estimate: Around 2.5–3.5, but potentially biased if state-specific trends violate parallel trends (true effect: 3.0)
Synthetic control estimate: Around 2.5–3.5, matching pre-treatment levels to construct the counterfactual
SDID estimate: Around 2.5–3.5 (true value: 3.0), typically closer to the truth than either DiD or SC alone
SDID unit weights: Sparse weights on control states — a few donors receive large weights, most receive near zero
SDID time weights: Concentrate on pre-treatment periods most similar to the post-treatment periods
Placebo-based inference: SDID standard error around 0.3–0.8, with the estimate significantly different from zero
Comparison: SDID should perform at least as well as whichever of DiD or SC is better in this setting
Panel dimensions: 30 states x 25 periods = 750 observations, with 1 treated state from period 16

Summary

In this lab you learned:

Standard DiD relies on parallel trends; synthetic control matches pre-treatment levels; SDID combines both approaches
SDID uses unit weights (to select comparable controls) and time weights (to select informative pre-periods)
In simulations, SDID is often at least as good as whichever of DiD or SC performs better
Inference for SDID uses placebo-based or jackknife standard errors, not conventional OLS standard errors
SDID is particularly useful when you have a single (or few) treated units and doubt that parallel trends hold exactly
The synthdid package in R provides a clean implementation; Python and Stata implementations are also available