Asymptotically Unbiased Synthetic Control Methods by Density Matching
Journal of Causal Inference.
- synthetic control
- comparative case studies
- endogeneity
- distributional effects
In one sentence. Standard synthetic control weights are biased because donor outcomes are correlated with the error term; matching distributions rather than fitting observed outcomes yields an asymptotically unbiased estimator and a full counterfactual distribution.
Where the bias comes from
Synthetic control methods predict the counterfactual outcome of a treated unit as a weighted combination of untreated units, with weights chosen to match the pre-treatment outcome path. The difficulty is an endogeneity problem: the outcomes of the untreated units used as regressors are correlated with the error term, so weights fitted to observed outcomes inherit a bias that does not vanish with more pre-treatment periods.
Matching distributions instead
The paper assumes that the outcome distribution of the treated unit can be approximated by a weighted mixture of the untreated units' distributions, and estimates the weights by matching that mixture to the treated unit's distribution rather than by fitting realized outcomes. Three consequences follow:
- The estimator is asymptotically unbiased under the stated assumptions.
- The mean squared error of the counterfactual prediction is reduced.
- The method returns the full distribution of the treatment effect, not only a point estimate — so quantile and tail effects are available, not just the mean.
Experiments confirm the behaviour implied by the theory.