Finite-Population Inference for Heterogeneity in Many-Group Synthetic Difference-in-Differences

Synthetic difference-in-differences is widely used to estimate treatment effects for many treated groups against a common donor pool. When the same donors are reused across groups, the group-specific estimates are cross-sectionally dependent, and plug-in second moments overstate effect heterogeneity. We develop finite-population inference for heterogeneity in many-group synthetic difference-in-differences: the projection of realized group effects on observed group covariates, the projected group-effect curve, the between-group variance, and the explained share. The theory combines a modular first-stage representation, a joint covariance kernel for donor sharing and block dependence, analytic and leave-out corrections for second moments, and calibrated omnibus and directed tests under explicit exchangeability or fit-matching conditions. In an American Community Survey application to the Affordable Care Act Medicaid expansion, whose estimand is the incremental effect of expansion status, pre-expansion uninsured rates explain much of the state-level effect variation on the percentage-point scale, household split-samples validate the decomposition, and donor sharing materially increases the standard error for the average effect. In a county-level Clean Air Act application, groupwise estimates are noisy, but a pre-specified projection on baseline fine-particulate pollution reveals a sign-stable directed component under state and division block covariance; placebo analyses attribute part of the raw gradient to regional convergence.