Synthetic Difference-in-Differences

Motivating Example

You are studying the effect of a state-level policy on employment. You have panel data for 50 states over 20 years. Ten states adopted the policy simultaneously.

DiD compares before-after changes in treated vs. control states, giving equal weight to all control states. But some control states are much more similar to the treated states than others.

Synthetic control constructs a weighted combination of control states to match each treated state's pre-treatment trajectory. But it is designed for a single treated unit and does not difference out common time effects the way DiD does.

What if you could get the strengths of both approaches?

Arkhangelsky et al. (2021) proposed Synthetic Difference-in-Differences (SDID), an estimator that combines unit weights (like synthetic control) with time weights (a novel addition) and includes an intercept shift (like DiD). The result is an estimator that tends to perform as well as or better than either component in a wide range of settings studied by the authors.

A. Overview

SDID works in three steps:

1. Unit weights ($\hat{\omega}$): Choose weights on control units so the weighted control group matches the treated group's pre-treatment outcome trajectory — like synthetic control.

2. Time weights ($\hat{\lambda}$): Choose weights on pre-treatment periods so the weighted pre-treatment average matches the post-treatment control average. This weighting handles the insight that recent pre-treatment periods may matter more.

3. Estimate the treatment effect using a doubly-weighted DiD-style regression.

When Does SDID Beat Its Components?

• If parallel trends is exactly right and all controls are equally good, DiD and SDID give similar results.
• If parallel trends fails but synthetic control matching is perfect, SC and SDID give similar results.
• In the common intermediate case, SDID tends to outperform both in the simulations of Arkhangelsky et al. (2021).

Common Confusions

"Is SDID just synthetic control with an intercept?" Close, but not quite. SDID adds both an intercept shift (like DiD) and time weights (which neither DiD nor SC uses). The time weights focus on the most predictive pre-treatment periods.

"Does SDID work with staggered treatment?" The original paper focuses on non-staggered settings. Extensions to staggered treatment are still developing. For staggered treatment, consider the staggered adoption DiD estimators such as Callaway and Sant'Anna (2021).

"When should I use SDID vs. plain DiD?" In the settings studied by Arkhangelsky et al. (2021), SDID tends to perform at least as well as DiD. The main cost is complexity and the requirement for enough pre-treatment periods to estimate meaningful weights. If parallel trends is highly plausible and you have many similar units, plain DiD may be sufficient and more transparent. Conducting a sensitivity analysis on the parallel trends assumption can help you decide.

B. Identification

The SDID Estimator

SDID solves:

The unit weights $\hat{\omega}_i$ solve:

The regularization $\zeta^2$ prevents overfitting, calibrated as $\zeta = (N_{tr} T_{post})^{1/4} \hat{\sigma}$.

The time weights $\hat{\lambda}_t$ are chosen analogously, matching pre-treatment control averages to post-treatment control averages.

Identifying Assumptions

1. No interference: One unit's treatment does not affect another unit's outcome.
2. No anticipation: Treatment has no effect before implementation.
3. Approximate parallel trends or approximate pre-treatment matching: SDID requires weaker conditions than either pure DiD or pure SC alone.

C. Visual Intuition

D. Mathematical Derivation

E. Implementation

1library(synthdid)
2
3# Prepare data as a matrix: Y is N x T, control units first
4# N0 = number of control units, T0 = number of pre-treatment periods
5
6# Step 1: Estimate SDID
7tau_sdid <- synthdid_estimate(Y, N0, T0)
8
9# Step 2: Compute standard error via placebo method
10# Reassigns treatment to each control unit and computes placebo effects
11se_sdid <- sqrt(vcov(tau_sdid, method = "placebo"))
12cat("SDID:", tau_sdid, "(SE:", round(se_sdid, 3), ")\n")
13
14# Step 3: Compare with DiD and SC for robustness
15# did_estimate uses equal weights (standard DiD)
16tau_did <- did_estimate(Y, N0, T0)
17# sc_estimate uses synthetic control weights (no intercept shift)
18tau_sc <- sc_estimate(Y, N0, T0)
19cat("DiD:", tau_did, "\nSC:", tau_sc, "\nSDID:", tau_sdid, "\n")
20
21# Step 4: Plot the SDID estimate with trajectory and weights
22plot(tau_sdid)

F. Diagnostics

1. Report unit weights. If approximately equal, SDID behaves like DiD. If concentrated, it behaves like SC.
2. Report time weights. Concentrated time weights indicate that recent pre-treatment periods dominate.
3. Pre-treatment fit. Compare weighted control trajectory to treated trajectory pre-treatment.
4. Compare DiD, SC, and SDID. Report all three. Agreement strengthens credibility.
5. Placebo inference. Reassign treatment to control units and re-estimate to construct standard errors. This approach is analogous to randomization inference in experimental settings.

Interpreting Your Results

All three estimators agree: Very strong evidence. Robust to different assumptions.

SDID and SC agree but DiD differs: Parallel trends may be violated; pre-treatment matching is doing the work.

SDID and DiD agree but SC differs: SC may be overfitting; SDID's regularization stabilizes the estimate.

G. What Can Go Wrong

H. Practice

Unit weights: top 5 of 30 control states receive 72% of weight
Time weights: 2012-2013 receive 65% of weight
Pre-treatment RMSPE: 0.42 (SDID), 0.38 (SC), 1.85 (DiD)

I. Swap-In: When to Use Something Else

• Canonical DiD: When parallel trends is credible without unit reweighting — standard DiD is simpler and may suffice.
• Synthetic control: When there is a single treated unit and the donor-pool approach is more transparent — SC is the original method on which SDiD builds.
• Staggered DiD: When treatment timing varies across units — SDiD as originally proposed handles simultaneous adoption more naturally than staggered settings.
• Event studies: When the full time profile of dynamic treatment effects is of primary interest rather than a single aggregate treatment effect.

J. Reviewer Checklist