3. Methodology

The primary objective of this paper is to examine whether the childbirth incentive program implemented by Incheon in 2024 —referred to as the "100 Million+ I-Dream"—has led to an improvement in the birth rate relative to other cities. If a positive effect is observed, the findings may offer valuable insights for future policy-making in the area of natal incentives. To address this research question, we employ the Synthetic Difference-in-Differences (SDID) method as our identification strategy. This approach exploits both temporal and geographical variation in birth rates across cities to estimate the causal impact of the policy.

 

In this section, we provide a concise overview of the SDID methodology and present both the theoretical and empirical support for its reliability in assessing the effect of Incheon's childbirth incentive policy on the birth rate.  

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3.1 Synthetic Difference in Differences

Evaluating the causal effects of policy interventions using panel data has been a central concern in empirical economics and social science research. In such settings, researchers often observe repeated measurements of units (e.g., cities, states) over time, where only some units are exposed to a policy during certain periods. A key challenge arises from the fact that policy adoption is typically non-random, leading to potential bias due to unobserved confounders that correlate with both treatment assignment and outcomes. These challenges complicate the task of drawing credible causal inferences from observational data.

To address these issues, a range of empirical strategies have been developed. Among them, Difference-in-Differences (DID) has become one of the most widely used approaches in applied economics over the last three decades. DID leverages variation across time and units to estimate causal effects, under the critical assumption that, absent treatment, the treated and control groups would have followed parallel trends. This method is particularly suited to settings with a substantial number of treated and untreated units, where additive unit and time fixed effects can sufficiently account for selection bias and common shocks.

More recently, the Synthetic Control (SC) method has emerged as a prominent alternative for settings with a single treated unit or a small number of treated units. SC constructs a weighted combination of untreated units that best replicates the pre-treatment trajectory of the treated unit, thereby improving pre-reatment balance and mitigating the need for a strict parallel trends assumption. As such, SC is well-suited for comparative case studies where traditional DID methods may perform poorly due to limited sample size or poor pre-treatment fit.

Although DID and SC are often applied in distinct empirical contexts, recent work has highlighted their theoretical connections. In particular, both can be viewed as special cases of two-way fixed effects models with different weighting and balancing structures. Building on this insight, Synthetic Difference-in-Differences (SDID), as proposed by Arkhangelsky et al., combines both methods. Like SC, SDID re-weights control units to achieve close pre-treatment balance with treated units, thus reducing dependence on the parallel trends assumption. Like DID, it incorporates unit and time fixed effects to control for unobserved heterogeneity and to improve estimation precision.

In addition to combining the strengths of DID and SC, the SDID introduces methodological innovations that enhance robustness and applicability. Arkhangelsky et al. demonstrate that SDID performs well both theoretically and empirically, particularly in settings with staggered treatment adoption or heterogeneous treatment effects. As such, SDID provides a flexible and credible approach for causal inference in observational panel data settings and serves as a useful tool in policy evaluation research.

 

Formally, the DID, SC, and SDID estimators recover the average treatment effect by solving following two-way fixed effects regression, respectively:

Equation (1)

Equation (2)

Equation (3)

Here, N denotes the total number of units, with N_co control units and N_tr treated units. T is the number of time periods, with T_pre indicating the number of pre-treatment periods. Y_it represents the outcome for unit i at time t, and W_it is a binary treatment indicator. In the SDID estimator, w^i_sdid are unit weights that align the pre-reatment trends of control units with those of treated units, while lambda^_t_sdid are time weights that balance pre- and post-treatment periods. Details on the construction of the unit and time weights w^ and lambda^ are provided in Appendix A. 

Unlike the standard DID estimator, SDID incorporates both unit and time weights, making the regression effectively “local.” It emphasizes observations—both across units and time periods—that are most similar to the treated group. This localization yields two key advantages: (1) improved robustness by focusing only on comparable units and time periods, and (2) enhanced precision through the implicit removal of systematic variation in outcomes attributable to unit or time heterogeneity. The unit weights are constructed such that the pre-treatment outcome trends of treated units are closely matched by a weighted average of control units. Similarly, time weights are chosen to ensure that post-treatment outcomes differ from pre-treatment trends by a constant shift, thereby improving temporal balance. Because raw panel data rarely exhibit parallel trends between treated and control units, SDID addresses this limitation through its weighting scheme. 

 

In contrast to the SC estimator—which omits unit fixed effects and time weights—SDID’s inclusion of both enhances robustness and precision. Specifically, time weights reduce bias by down-weighting periods dissimilar to the treatment window, while unit fixed effects flexibly control for persistent, unobserved heterogeneity across units, often capturing substantial variation in the outcome variable.

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3.2 Empirical Specifications

The objective of this analysis is to estimate the impact of Incheon’s “100 Million+ I-Dream” program on the fertility rate and the number of births. We use panel data covering 16 cities—including Incheon—from 2000 to 2024. The policy was implemented in Incheon in 2024, yielding T_pre = 24 pre-treatment periods and T_post = T - T_pre - 1 post-treatment period, with N_tr = 1 treated unit (Incheon) and N_co = 15 control units.

We first estimate Equation (3) without control variables (Section 4.1), and then re-estimate the same equation incorporating control variables, based on the method proposed by Arkhangelsky et al. (2021) (Section 4.2).

 

It is important to note that since our empirical strategy is not confined to estimating a single-time effect or the average effect across all post-treatment periods. Instead, it allows for heterogeneity in treatment effects over time. Therefore, the findings of this study are well-positioned to be extended to future periods. This enables us to capture and analyze the long-term, staggered effects of the policy on birth rates. In future work, we will conduct a follow-up analyses to examine how the policy's effects evolve over time.

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