1 Introduction

Two of the most widely used methods for evaluating policies and interventions in social science are difference-in-differences (DiD) and the synthetic control method (SC). Both use control units to construct a counterfactual for a treated unit. Both require an assumption about the absence of treatment in the control group. Yet they are built on different foundations, make different assumptions, and suit different research designs.

The choice between them is consequential. Applying DiD when SC is more appropriate- or vice versa can yield badly misleading estimates. This article steelmans both approaches, identifies the conditions under which each is preferred, and discusses hybrid methods that combine their strengths.

2 The Case for Difference-in-Differences

2.1 Simplicity and Transparency

DiD has a compelling simplicity: it subtracts the pre-post change in control units from the pre-post change in treated units. The estimator is:

DiD (YT.post-YT.pre) (Yc.postYc.pre)

(1)

This can be estimated by OLS and extended to multiple time periods, multiple groups, and covariates straightforwardly. The identifying assumption parallel trends is transparent and, to some degree, empirically testable via pre-trend tests.

2.2 Multiple Treated Units

DiD shines when there are many treated units. With a large treatment group, the DiD estimator has low variance, and heterogeneity across treated units can be studied. The modern staggered DiD literature Callaway-Sant'Anna, Goodman-Bacon, de Chaisemartin- D'Haultfoeuille is built on DiD and provides tools for settings with staggered treatment timing across many units [Roth et al., 2023].

SC, in contrast, was designed for a single (or very few) treated units. With many treated units, SC is computationally cumbersome and requires solving a separate optimisation for each treated unit.

2.3 Inference

DiD inference is straightforward under standard panel regression: cluster standard errors at the unit level, or use block bootstrap. Asymptotic theory is well-developed for large N.

SC inference via placebo permutations is valid but has low power when the donor pool is small. With fewer than 40-50 donor units, the permutation distribution is coarse, and meaningful p-values are hard to achieve.

3 The Case for Synthetic Control

3.1 One (or Few) Treated Units

The most important advantage of SC is its suitability for comparative case studies with a single treated unit a country, state, or city. DiD with a single treated unit relies on a single control or an unweighted average of controls, both of which may be poor comparators. SC constructs a weighted comparison that closely matches the treated unit in the pre-treatment period, reducing pre-treatment bias by design.

As Abadie [2021] argues, when the treated unit is unique California's tobacco programme, Germany's reunification, an individual country's trade reform-SC is more appropriate than DiD because there is no natural comparison unit.

3.2 Transparent Pre-Treatment Fit

A critical advantage of SC over DiD is that the pre-treatment fit is directly observable. The researcher can inspect how closely the synthetic control tracks the treated unit before treatment and report the pre-treatment MSPE (mean squared prediction error). Poor pre-treatment fit is a visible warning sign that the counterfactual is unreliable.

In DiD, the parallel trends assumption concerns the counterfactual trend post-treatment, which is unobservable. The pre-trend test is an imperfect proxy. SC makes the analogous check much more transparent.

3.3 Handling Non-Parallel Trends

DiD requires that treated and control units would have followed parallel trends in the absence of treatment. SC relaxes this: it allows treated and control units to have different pre-treatment levels and trends, as long as a weighted combination of control units can replicate the treated unit's pre-treatment path. This is a weaker assumption when there are systematic differences between the treated unit and any single control.

4 Key Differences

5 The Hybrid Approach: Synthetic DiD

A compelling synthesis is Synthetic DiD (SDiD) proposed by Arkhangelsky et al. [2021]. SDiD constructs unit weights (as in SC) to remove pre-treatment level differences and time weights to down-weight early pre-treatment periods mimicking the "local" comparison of DiD.

The estimator satisfies a double robustness property: it is consistent if either the unit weights or the time weights are correctly specified.

SDiD handles multiple treated units more naturally than SC, and it outperforms standard DiD when control units have heterogeneous trends. It is implemented in the synthdid R package.

6 A Decision Framework

Researchers facing a design choice should consider:

1. How many treated units?

• One or two: Synthetic control (or SDiD).

• Many (e.g., 20+ treated states): DiD (possibly staggered).

• 3-15: Both are viable; SDiD is a strong default.

2. Is parallel trends plausible without weighting?

• If the treated unit is visibly different in pre-treatment levels: SC or SDiD.

• If treated and control units look similar pre-treatment: DiD is efficient and straightforward.

3. Is the treated unit within the convex hull of control units?

• If the treated unit is an extreme case (highest income country, largest city): SC extrapolates poorly. Use DiD or ASCM.

4. Are treatment effects dynamic (growing or decaying)?

• Both methods can accommodate dynamics; event-study plots are informative for both.

7 An Empirical Illustration: The Mariel Boatlift

The Mariel Boatlift controversy illustrates the sensitivity of the method choice. Card [1990] used DiD with a single treated city (Miami) and a comparison group of four cities. Borjas [2017] re-analysed the data with different control groups and found larger wage effects. Peri and Yasenov [2019] used SC to construct an optimal synthetic Miami and found effects closer to Card's original null result.

The disagreement turned partly on which comparisons are most valid exactly the question that SC and DiD answer differently. SC's data-driven weighting produced a synthetic Miami with much better pre-treatment fit than the simple DiD comparator, suggesting the SC result is more credible in this case [Abadie, 2021].

8 Conclusion

DiD and SC are complements, not substitutes. DiD is the method of choice when there are many treated units, parallel trends is plausible, and a rich pre-treatment period is available to test for trend differences. SC is the method of choice for comparative case studies with one or few treated units, where constructing a data-driven counterfactual improves on any natural comparison. Synthetic DiD combines the strengths of both and is an attractive default in intermediate settings. The key is to choose the method based on the research design not based on which produces the preferred result.

References

1. Abadie, A., Diamond, A., and Hainmueller, J. (2010). Synthetic control methods for comparative case studies. Journal of the American Statistical Association, 105(490):493-505.
2. Abadie, A. (2021). Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature, 59(2):391-425.
3. Arkhangelsky, D., Athey, S., Hirshberg, D.A., Imbens, G.W., and Wager, S. (2021). Synthetic difference-in-differences. American Economic Review, 111(12):4088-4118.
4. Card, D. (1990). The impact of the Mariel boatlift on the Miami labor market. Industrial and Labor Relations Review, 43(2):245-257.
5. Borjas, G.J. (2017). The wage impact of the Marielitos: A reappraisal. ILR Review, 70(5):1077-1110.
6. Peri, G. and Yasenov, V. (2019). The labor market effects of a refugee wave: Synthetic control method meets the Mariel boatlift. Journal of Human Resources, 54(2):267-309.
7. Roth, J., Sant'Anna, P.H.C., Bilinski, A., and Poe, J. (2023). What's trending in difference-in-differences? A synthesis of the recent econometrics literature. Journal of Econometrics, 235(2):2218-2244.