On the sparsity of synthetic control method

ABSTRACT

Synthetic Control Method (SCM) is a popular approach for causal inference in panel data, where the optimal weights for control units are often sparse. But the sparsity of SCM has received little attention in the literature except Abadie (2021), which explores the sparsity from the perspective of predictor space. In this paper, we make three contributions. First, we show that if there is a unique solution, then the number of positive weights is upper-bounded by the number of covariates. Second, we offer a simple alternative explanation about the sparsity of SCM from the perspective of parameter space. Third, we conduct a meta-analysis of empirical studies using SCM in the literature, which shows that the sparsity of SCM decreases with the relative number of covariates. A practical implication is that if the number of positive weights exceeds the number of covariates, there are multiple solutions and possibly unstable weights.

KEYWORDS:

• Synthetic control method
• sparsity
• meta-analysis

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ In this paper, we use the terms “covariate” and “predictor” interchangeably.

² One may regress among $\left\{{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_{\mathit{N}+1}\right\}$ and drop the variable with the highest $R^2$, and so on.

³ For illustration purpose, we do not use the options “nested allopt” for more accurate numerical computation, otherwise the algorithm would fail when there is only one covariate (K = 1).

⁴ Note that Abadie et al. (2010) never estimates the one-covariate case.

⁵ Among these 30 studies with all positive weights, the number of covariates ranges from 3 to 14 with a mean of 9, and the number of control units ranges from 4 to 54 with a mean of 17.43.

⁶ See Appendix 2 for a complete list of these papers used in the meta-analysis.

⁷ Note that all 61 papers with multiple SCM studies used the same donor pools throughout, and 49 of them used the same covariates as well, but 12 of them used different covariates due to missing values.

⁸ The paper fixed effects term $u_i$ captures unobserved heterogeneity across papers, which is analogous to the individual fixed effects in the panel data setting. Since $u_i$ may be correlated with the regressors, its omission could result in inconsistent estimation. Moreover, since some paper dummies are significant in the regressions (unreported), the paper fixed effects specification is preferred over the pooled OLS.

⁹ The opposite specific-to-general approach starts from the smallest model possible (i.e., a model with only the constant term), and proceeds by adding the most significant regressor at each step. However, the specific-to-general approach may suffer from omitted variable bias when the model is too small.

Additional information

Notes on contributors

Qiang Chen

Qiang Chen is a professor at the School of Economics, Shandong University.

Wenjun Li

Wenjun Li is a PhD student at the School of Economics, Shandong University.