Synthetic Difference-in-Differences - Index

Routing Summary

Two linked responses to fragile parallel trends in panel causal inference. (A) Construct parallel trends: Synthetic Difference-in-Differences (Arkhangelsky, Athey, Hirshberg, Imbens & Wager 2021, AER) — a TWFE regression weighted by synthetic-control-style unit weights and novel time weights. (B) Interrogate parallel trends: event-study designs, Roth’s (2022, AER: Insights) critique of pre-trend testing, and Rambachan & Roth’s (2023, REStud) Honest DiD sensitivity analysis, with Roth, Sant’Anna, Bilinski & Poe (2023) as the synthesising survey. Fills Dream gaps #47 and #45.

• Need the big picture / where SDID sits between DiD and SC? → Synthetic Difference-in-Differences - Overview
• Need the optimisation problems for $\hat{\omega }$, $\hat{\lambda }$, $\zeta$ and Algorithm 1? → SDID Estimator - Unit and Time Weights
• Need why it works (factor model, bias decomposition, double robustness) and the CPS / Penn World Table simulation evidence? → SDID vs DiD vs Synthetic Control
• Need standard errors — especially with a single treated unit? → SDID Inference - Bootstrap, Jackknife and Placebo
• Need to apply it to a geo test, matched-market test or marketing panel? → SDID for Geo Experiments and Marketing Panels
• Need the leads-and-lags regression, normalisation, anticipation, binning, staggered-timing contamination? → Event Study Designs and Dynamic Treatment Effects
• Need to know why “no significant pre-trend” is weak evidence (power, pre-test bias)? → Pre-Trend Testing and Its Pitfalls
• Need robust confidence sets / a breakdown value when parallel trends may fail? → Honest DiD - Sensitivity to Parallel Trends Violations
• Need the headline asymptotic result? → Theorem 1
• Need the double-robustness identity? → bias decomposition

Concept Map

Concept	Note	Type	Depends On	Key Result
SDID framing; DiD/SC/SDID as one regression family	Synthetic Difference-in-Differences - Overview	overview	DiD; Synthetic Control; Fixed-Effects Model	${\hat{\tau }}^{sdid}$ = TWFE weighted by ${\hat{\omega }}_i{\hat{\lambda }}_t$; Prop 99: SDID $-15.6(8.4)$ vs SC $-19.6(9.9)$ vs DID $-27.3(17.7)$
Unit weights, time weights, regularisation	SDID Estimator - Unit and Time Weights	method	Overview; Synthetic Control	Eqs. 2.1-2.3; $\zeta =(N_{tr}{T}_{post}{)}^{1/4}\hat{\sigma }$; weighted double-difference form (4.3); invariance to ${\alpha }_i+{\beta }_t$ shifts
Factor-model bias analysis and simulation evidence	SDID vs DiD vs Synthetic Control	concept	Estimator; SC Bias Theory	$B(\omega ,\lambda )$ vanishes if either unit or time weights balance $L$; SDID RMSE 0.028 vs DID 0.049 (CPS), 0.031 vs 0.197 (PWT)
Large-sample inference	SDID Inference - Bootstrap, Jackknife and Placebo	method	Estimator; Standard Errors and Clustering	Thm 1 asymptotic normality with oracle variance; Thm 2 jackknife conservative; placebo needs homoskedasticity, only option for $N_{tr}=1$
Application to geo lift and marketing panels	SDID for Geo Experiments and Marketing Panels	application	All SDID notes; Geo-Experiment Methodology	Non-random market selection biases DiD; SDID about 2× more precise even under randomisation; practical checklist
Event-study specification and dynamic effects	Event Study Designs and Dynamic Treatment Effects	concept	DiD; Staggered DiD assumptions	$\beta =\tau +\delta$, ${\tau }_{pre}=0$; coefficients are 2×2 DiDs sharing reference-period noise; dynamic TWFE contaminated under staggered heterogeneity
Limits of pre-trend tests	Pre-Trend Testing and Its Pitfalls	concept	Event Study	Prop. 1 pre-test bias $={\Sigma }_{12}{\Sigma }_{22}^{-1}(\mathbb{E}[{\hat{\beta }}_{pre}\mid \text{pass}]-{\beta }_{pre})$; Prop. 2 exacerbation under monotone trends; rejection rates up to 0.98
Partial identification under bounded violations	Honest DiD - Sensitivity to Parallel Trends Violations	method	Event Study; Pre-Trend Testing	${\Delta }^{RM}(\bar{M})$, ${\Delta }^{SD}(M)$; identified set = estimate − worst-case bias; hybrid/FLCI confidence sets; breakdown value

Notes

• Synthetic Difference-in-Differences - Overview — CONTAINS: block-assignment setting and estimand, estimator family table (DiD / SC / DIFP / SDID), headline results, cluster map, marketing-measurement relevance, Prop 99 table, synthdid usage.
• SDID Estimator - Unit and Time Weights — CONTAINS: eq. 2.1 unit weights with intercept and ridge, eq. 2.2 $\zeta$, eq. 2.3 time weights, Algorithm 1, covariate residualisation, weighted double-differencing (4.3) and adjusted outcomes ${\hat{\delta }}_i$ (2.4-2.5), invariance property, autoregression vector $\psi$, staggered-adoption recipe, California weight tables, CVXPY sketch.
• SDID vs DiD vs Synthetic Control — CONTAINS: latent factor model (3.2/4.1), error decomposition (4.4), double-robustness identity, oracle weights and three-term error (4.8), untestable-assumption caveat, comparison with IFE/GSC/matrix completion and augmented SC (6.1), CPS and Penn World Table placebo tables, rules of thumb.
• SDID Inference - Bootstrap, Jackknife and Placebo — CONTAINS: Assumptions 1-4, Theorem 1, Algorithms 2-4, Theorem 2, placebo-method assumptions and relation to Conley–Taber and randomisation inference, method comparison table, Table 4 coverage, jackknife code sketch.
• SDID for Geo Experiments and Marketing Panels — CONTAINS: mapping of GBR / TBR / CausalImpact / augmented SC to the panel-estimator taxonomy, geo test as SDID panel and iROAS, three lessons from the paper’s empirical studies, 9-point practical checklist, when not to use SDID, R workflow.
• Event Study Designs and Dynamic Treatment Effects — CONTAINS: dynamic TWFE specification, 2×2 equivalence and induced correlation, causal decomposition, normalisation / anticipation / baseline / binning / simultaneous-band conventions, Sun–Abraham contamination result, $AT{T}_l^w$ aggregation, Python sketch.
• Pre-Trend Testing and Its Pitfalls — CONTAINS: four problems with pre-testing, normal model and NIS test, survey of 12 papers, power calibration (${\gamma }_{0.5},{\gamma }_{0.8}$), Propositions 1-4, publication-filter model (eq. 4), recommendations, verified simulation of pre-test bias.
• Honest DiD - Sensitivity to Parallel Trends Violations — CONTAINS: identified set and Lemma 2.1, ${\Delta }^{RM}$ / ${\Delta }^{SD}$ / ${\Delta }^{SDRM}$ / sign / polyhedral classes, uniform coverage criterion, ARP conditional and hybrid tests, optimal FLCI with Props. 4.1-4.2, choice-of-$\Delta$ guidance, breakdown values, two empirical illustrations, HonestDiD workflow.

External / Cross-Folder Links

• Differences-in-Differences, Bayesian Difference in Differences, Fixed-Effects Model — parent design and regression.
• Synthetic Control, Synthetic Control Bias Theory, Synthetic Control Requirements, Synthetic Control Inference and Diagnostics, Synthetic Control Extensions, Abadie 2021 - Overview — the SC cluster SDID builds on.
• Generalized Synthetic Control Method, Xu 2016 - Overview — explicit interactive-fixed-effects alternative.
• Difference-in-Differences with Multiple Time Periods - Overview, Group-Time Average Treatment Effects, Identifying Assumptions for Staggered DiD, Aggregating Group-Time Effects, Doubly-Robust Estimands for ATT(g,t), Simultaneous Inference via Multiplier Bootstrap — staggered DiD cluster.
• Standard Errors and Clustering, Randomization Inference - Overview, Permutation Tests and Exact Inference, Fisher Randomization Test and the Sharp Null — inference foundations.
• Geo-Experiment Methodology - Overview, Geo-Experiment Design and Power Analysis, Time-Based Regression Estimator for Geo Experiments, TBR Design Sensitivity and the Stationarity Assumption, Counterfactual Impact Estimation — marketing-measurement counterparts.
• Sensitivity Analysis in Observational Studies, Plausible GMM - Overview — sensitivity / partial-identification relatives.
• Carryover Effects and Distributed Lags — dynamic effects as lag distributions.

Sources

• Arkhangelsky 2021 - Synthetic Difference in Differences — Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W. & Wager, S. (2021), “Synthetic Difference-in-Differences,” American Economic Review 111(12). arXiv:1812.09970v4.
• Roth 2022 - Pretest with Caution — Roth, J. (2022), “Pretest with Caution: Event-Study Estimates after Testing for Parallel Trends,” AER: Insights 4(3): 305–322.
• Rambachan Roth 2023 - A More Credible Approach to Parallel Trends — Rambachan, A. & Roth, J. (2023), “A More Credible Approach to Parallel Trends,” Review of Economic Studies 90(5): 2555–2591.
• Roth 2023 - Whats Trending in Difference-in-Differences — Roth, J., Sant’Anna, P. H. C., Bilinski, A. & Poe, J. (2023), “What’s Trending in Difference-in-Differences? A Synthesis of the Recent Econometrics Literature,” Journal of Econometrics. arXiv:2201.01194.