Synthetic Difference In Differences

Synthetic Control (SC) methods, popular for estimating treatment effects in panel data, use data-driven weights to balance pre-treatment outcomes for treated and control units. This paper offers a novel perspective on SC, representing it as a weighted least squares regression with time fixed effects. Building on this, the authors propose a significant improvement: the Synthetic Difference-in-Differences (SDID) estimator. SDID incorporates both unit and time fixed effects, along with unit and time weights, creating a doubly weighted local version of the standard DID estimator. This enhancement addresses potential bias by allowing for flexibility and more closely resembling the period(s) for which counterfactuals are imputed. The paper contrasts SDID with both SC and DID, highlighting its superior bias properties and double robustness. Unlike SC and DID, SDID's consistency is ensured if either the model is correctly specified or the weights are well-chosen, without needing both. The introduction concludes by mentioning various existing approaches to SC, such as those focusing on balancing weights or modeling conditional outcomes, and highlighting that SDID uniquely offers a weighted regression characterization, easily accommodating covariates and generalizing to multiple treated units and periods.

The paper reviews existing synthetic control methods, categorizing them into those focusing on balancing weights and those focusing on modeling conditional outcomes. It highlights the work of Abadie and co-authors on synthetic control methods, emphasizing their seminal contributions to the field. The authors mention the work of Doudchenko and Imbens on characterizing synthetic control weights, and the contributions of Xu, Athey et al., Carvalho et al., Li and Bell on various aspects of synthetic control and related methods. The paper particularly notes the exception of Ben-Michael, Feller, and Rothstein's Augmented Synthetic Control (ASC) estimator, which combines outcome modeling with balancing weights, resulting in double robustness. However, the paper emphasizes that ASC lacks a weighted regression characterization, unlike SDID. The authors further contextualize their approach within the broader program evaluation literature, citing works that combine weighting/balancing with outcome modeling and their associated double robustness properties. This literature review positions SDID as a novel approach that synthesizes these existing methodologies, offering both theoretical improvements and practical advantages.

The paper's core methodology centers on the development and analysis of the SDID estimator. It begins by characterizing the standard SC estimator as a weighted least squares regression with time fixed effects and unit-specific weights. The authors then introduce SDID, which extends this representation by adding unit fixed effects and time weights. The weights in SDID are crucial; they can be chosen using various methods, including those inspired by the SC approach (balancing pre-treatment outcomes), or methods that emphasize recent periods. The paper explores different weighting schemes, such as SC weights (with L2 penalization to avoid overfitting), time-equivalent SC weights, intercept weights to account for trends, kernel weights, and nearest neighbor weights. The methodology section details the estimation procedure, explaining how the SDID estimator is obtained by minimizing a weighted least squares objective function that includes both unit and time fixed effects. The theoretical justification relies on asymptotic analysis (large N and large T) assuming that the outcome variable is a noisy estimate of an underlying signal matrix. The paper establishes consistency results under different assumptions, highlighting the double robustness of SDID. One result emphasizes weak assumptions on the weights and strong assumptions on the outcome model, while another emphasizes the opposite. The authors use tools from empirical risk minimization theory to demonstrate that despite being optimized on observed data, the weights effectively balance the underlying signal, even with a large number of units and time periods. The methodology also includes details on how to perform inference using robust standard errors from weighted DID regressions, treating the weights as fixed despite their data dependence.

The paper's key findings revolve around the superior properties of the SDID estimator. Theoretically, the authors prove that SDID offers double robustness. Specifically, under a well-specified two-way fixed effects model, SDID is consistent under flexible conditions on the weights. Furthermore, even when the fixed effects model is misspecified and the data follow a low-rank structure, SDID remains consistent with appropriately penalized SC weights. The paper also proves consistency results for the original SC estimator under a low-rank assumption on the underlying data-generating process, but notes that the conditions on the weights are stronger than for SDID. The authors establish conditions under which standard robust standard errors for weighted DID regressions are valid, even when the weights depend on the data. Empirically, the paper presents results from a placebo study on smoking prevalence data and simulation studies. The placebo study, replicating and extending the original California tobacco control program study, shows that SDID substantially outperforms both SC and DID in terms of predictive accuracy. Simulation results across various settings confirm SDID's superior bias properties and often better root-mean-squared-error compared to DID and SC. The simulations also demonstrate the effectiveness of SDID's weighted regression perspective for inference, producing well-calibrated confidence intervals that outperform those from a basic DID approach, even with correlated errors. The study shows that jackknife-based inference for SDID is robust to within-row error correlations in the data.

The findings address the research question by demonstrating that SDID is a superior estimator for causal effects in panel data settings compared to existing methods. The double robustness of SDID provides a significant advantage, making it less sensitive to model misspecification. This is crucial in settings where the underlying data-generating process is complex and may not perfectly fit a simple two-way fixed effects model. The superior empirical performance of SDID, shown through the placebo study and simulations, corroborates the theoretical findings. The results underscore the importance of combining both weighting/balancing and outcome modeling for improved causal inference. The applicability of standard inference methods further enhances the practicality and usability of SDID. The findings are relevant to various fields employing panel data analysis, where accurate estimation of treatment effects is critical. The method's robustness to correlated errors and adaptability to different sampling assumptions broaden its potential applications and improve its overall reliability.

This paper introduces the SDID estimator, providing a significant advance in causal inference for panel data. SDID combines the strengths of synthetic control and difference-in-differences methods, resulting in an estimator with superior bias properties and double robustness. The paper's theoretical results and empirical evaluations confirm SDID's advantages over existing methods. Future research could explore more sophisticated weighting schemes or investigate extensions to settings with more complex treatment patterns or heterogeneous treatment effects. Further investigation into optimal methods for variance estimation, especially in high-dimensional settings, would also be valuable.

The paper's reliance on asymptotic results implies that the finite-sample properties of SDID might deviate from the theoretical predictions in smaller samples. The performance of SDID depends critically on the choice of weights, and optimal weight selection methods may vary across applications. The assumption of a low-rank structure for the underlying signal matrix in the data-generating process might be restrictive in some applications, and relaxing this assumption could be a direction for future research. Although the paper addresses correlated errors, the specific types of correlation addressed might not encompass all possible scenarios encountered in practice.