Shared-Donor Inference for Heterogeneity in Many-Group Synthetic Difference-in-Differences

Many policy studies estimate separate synthetic-control or synthetic difference-indifferences effects for several treated groups and then summarize their heterogeneity. Reusing donors makes the estimated effects jointly dependent, and plug-in dispersion also contains first-stage estimation noise. Starting from a joint first-stage representation, we derive three results. The first propagates the shared-donor covariance to finite-set means, projections, contrasts, and projected effect curves. The second gives an exact analytic trace correction for total and explained heterogeneity and a high-level Gaussian limit for regular quadratic targets. Feasible many-block inference additionally requires consistent estimation of the corresponding limiting covariance. The third result concerns a fixed treated set: when sampling-center heterogeneity is zero, the linear approximation degenerates and a bootstrap that reproduces the first-order effectvector law yields quadratic boundary inference. Persistent counterfactual mismatch is reported separately through deterministic sensitivity calculations. In an American Community Survey analysis of Medicaid expansion, the full covariance increases the standard error of the mean effect from 0.256 to 0.456 percentage points, while the centered baseline uninsured-rate slope changes little. Trace correction also removes a nonnegligible part of the raw cross-state dispersion.