Preference rationalization for weight selection under Synthetic Parallel Trends

Determine preference structures over weighting schemes for control units (for example, penalties on the ℓ1 or ℓ2 norm, sparsity, or other utility-based criteria) that rationalize the selection of a particular counterfactual value for the treated unit within the identified set M under the Synthetic Parallel Trends assumption, by formally characterizing the mapping from policymaker preferences over weights ω to the resulting chosen counterfactual outcome μ_T^1(0).

Background

The paper shows that many popular panel-data methods, including difference-in-differences and synthetic control, can be understood as selecting specific population weights ω that reproduce the treated unit’s pre-treatment trends and then extrapolate to the post-treatment period. Under the Synthetic Parallel Trends (SPT) assumption, the counterfactual is set-identified because multiple weights may satisfy the pre-trend balancing, yielding an identified set M comprised of different counterfactual values.

In the remark on penalized regressions, the author connects common regularization choices (e.g., LASSO, ridge) to implicit preferences over weighting schemes. The open question is to reverse this logic: given a chosen counterfactual value in M that a policymaker prefers, what explicit preference over weights would justify that choice? This would provide a decision-theoretic foundation for selecting among multiple valid weights in SPT.