Introduction
For much of the past quarter-century, difference-in-differences (DiD) and the synthetic control method have evolved along parallel tracks—two distinct answers to the same question of constructing a counterfactual for a treated unit in panel data. DiD imposes parallel trends: absent treatment, the treated unit's outcome would have moved in step with a comparison group. Synthetic control imposes a factor model: the treated unit's counterfactual can be approximated by a weighted combination of donors whose pre-treatment path matches the treated unit's trajectory.
Each assumption has well-known failure modes. Parallel trends can fail when treated and control units are on divergent pre-treatment trends driven by unobservable factors. The synthetic control convex-hull constraint can fail when the treated unit lies outside the support of donor-pool outcomes, requiring extrapolation. A natural question, asked with increasing urgency since the synthetic DiD (SDiD) of Arkhangelsky et al. [2021] and the augmented synthetic control (ASCM) of Ben-Michael et al. [2021], is: can we combine the two assumptions so that failure of one does not sink the entire analysis?
A 2025 working paper by Sun et al. [2025] answers yes, providing the first formally doubly robust estimator that combines DiD parallel trends with synthetic control conditions. This article explains the logic, the estimator, its assumptions, and why it matters for applied practice.
1 Two Assumptions, One Estimator
1.1 The Setup
Consider a balanced panel with N units observed for T periods. Unit 1 receives treatment beginning in period T₀+1 and is the sole treated unit; units 2, ..., N are untreated donors. Let Yᵢₜ(0) denote the potential untreated outcome and Yᵢₜ(1) the potential treated outcome. The average treatment effect on the treated in post-period t > T₀ is:
The fundamental challenge is that Y₁ₜ(0) is unobserved after treatment. Both DiD and synthetic control propose specific models for this counterfactual.
1.2 The DiD Assumption
Standard DiD assumes that:
meaning that the change in the treated unit's counterfactual outcome equals the average change in a comparison group. This imposes parallel trends but allows the treated unit's level to be arbitrary.
1.3 The Synthetic Control Assumption
The synthetic control method assumes that there exist donor weights ωᵢ ≥ 0, Σᵢ≥₂ ωᵢ = 1, such that the weighted donor average closely approximates the treated unit's pre-treatment trajectory:
The identifying assumption is that this pre-treatment fit carries over to the post-treatment period.
1.4 The Doubly Robust Hybrid
Sun et al. [2025] propose an identification strategy that requires only one of the two assumptions to hold—a doubly robust structure analogous to the doubly robust DiD of Sant'Anna and Zhao [2020]. Their estimator has the form:
where ω̂ are synthetic control weights estimated from pre-treatment data and Δ̂ₜˢᶜ is a bias-correction term derived from the SC model. When parallel trends holds exactly, the SC correction has expectation zero and the estimator reduces to a DiD. When synthetic control holds exactly, the DiD component is bias-free and the SC correction is informative. Under either assumption alone, the estimator is consistent.
The key insight is that the two assumptions are mutually protective: an analyst who is uncertain whether the trends assumption or the SC factor model is the more appropriate representation of the data need not choose. Consistency obtains under the union of the two.
2 The Efficiency Frontier
Beyond double robustness, Sun et al. [2025] establish efficiency results. Under a semiparametric model, their estimator attains the efficiency bound when both assumptions hold simultaneously. When only one assumption holds, the estimator remains consistent but may not be fully efficient.
This is analogous to doubly robust estimation in cross-sectional settings [Robins et al., 1994]: a correctly specified outcome model and a correctly specified propensity score each separately ensure consistency, while both being correct yields the semiparametric efficiency bound.
The efficiency gain over plain DiD when both assumptions hold reflects the additional information contained in the pre-treatment fit of the synthetic control. Conversely, the efficiency gain over plain SC reflects the additional discipline imposed by parallel trends, which regularises the SC weights toward equal weighting.
3 Relationship to Existing Hybrid Methods
The taxonomy of hybrid DiD/SC estimators has grown considerably since 2020. Table 1 places the main proposals in context.
Synthetic DiD (SDiD) of Arkhangelsky et al. [2021] reweights both units and time periods to align pre-treatment trends. It shares the flavour of doubly robust estimation but does not formally satisfy the doubly robust property.
ASCM of Ben-Michael et al. [2021] augments the SC estimate with an outcome model correction term. It reduces bias when the SC weights do not perfectly match the pre-treatment path, but identification rests primarily on the SC factor model.
DR-DiD of Sant'Anna and Zhao [2020] is doubly robust in the sense of combining an outcome regression model with an inverse probability weighting model. However, its identification rests entirely on conditional parallel trends and does not invoke SC factor model assumptions.
The Sun-Xie-Zhang estimator is the first proposal to combine both the parallel trends assumption and the SC factor model assumption in a formally doubly robust framework.
4 Practical Guidance
4.1 When Is the Hybrid Most Valuable?
The doubly robust hybrid is most valuable when an applied researcher is uncertain between two credible but mutually incompatible identification stories. Consider a researcher studying the effect of a state-level policy reform:
- If the treated state has a pre-treatment trend very similar to control states, parallel trends looks plausible. But the researcher is unsure whether unobservable trends might diverge post-reform.
- If the treated state's pre-treatment outcome path can be well approximated by a weighted combination of donors, SC looks plausible. But the researcher is unsure whether the weights will remain stable post-reform.
In this scenario, the hybrid provides robustness: if either concern materialises, the estimate remains consistent.
4.2 Pre-Treatment Fit and Diagnostic Tests
For the SC component, pre-treatment mean squared prediction error (MSPE) should be minimised. For the DiD component, event-study pre-trend tests [Roth, 2022] should show no significant pre-trends. If both diagnostics look good, confidence in the hybrid estimate is high. If the SC pre-treatment fit is poor but pre-trends pass, the estimate reduces approximately to a DiD. If pre-trends fail but SC pre-treatment fit is good, the estimate leans on the SC assumption.
4.3 Inference
Sun et al. [2025] propose a permutation-based inference procedure that exploits the panel structure: placebo estimators are computed by treating each donor unit as if it had been treated, providing a null distribution for the test statistic. This extends the permutation inference of Abadie et al. [2010] to the hybrid estimator.
5 Connections to Staggered Settings
The Sun-Xie-Zhang framework is developed for a single treated unit, but the staggered adoption setting is both more common and more complex. Extending doubly robust DiD/SC hybrids to staggered adoption requires combining the group-time ATT logic of Callaway and Sant'Anna [2021] with SC weight construction for each cohort.
This is an active research frontier. Ben-Michael et al. [2021] extend ASCM to staggered settings using a within-cohort SC weight construction, and Arkhangelsky et al. [2021] extend SDiD to staggered designs. A fully doubly robust staggered hybrid remains an open problem, though the ingredients—doubly robust staggered DiD from Callaway and Sant'Anna [2021] and Sant'Anna and Zhao [2020] combined with SC—are available.
6 Conclusion
The doubly robust DiD/SC hybrid represents a significant methodological advance for settings where researchers are uncertain between two plausible identifying assumptions. By combining parallel trends and the synthetic control factor model in a single estimator, Sun et al. [2025] provide insurance against the failure of either assumption individually. This extends to the DiD/SC context a principle that has proved enormously valuable in cross-sectional causal inference: doubly robust estimation protects against model misspecification by hedging two complementary identifying conditions.
For applied researchers, the practical implication is clear: when you have a setting where both parallel trends and synthetic control pre-treatment fit look credible, the hybrid estimator provides efficiency gains over either method alone and robustness against the failure of one. When only one assumption is credible, the hybrid reduces to the appropriate single-assumption estimator.
References
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- Sun, L., Xie, Y., and Zhang, Y. (2025). Difference-in-differences meets synthetic control: A doubly robust approach. arXiv preprint arXiv:2503.11375.