On the sparsity of synthetic control method

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.