Beyond the Oracle: Inference with Estimated Groups

Models with latent groups are estimated in two steps: a classifier assigns units to groups, and estimated memberships are treated as data. Standard inference relies on the oracle property, which treats classification error as negligible, an assumption that is dubious in practice. I develop a local misclassification framework in which classification error and sampling noise are of the same order, and derive the first-order bias in linear models with grouped fixed-effects. Three findings matter most for practice: misclassification does not attenuate but mixes group-specific distortions, with weights of either sign; residual-based classifiers endogenously offset a fraction of the bias; and the time dimension attenuates bias when group paths are unsynchronized, beyond its effect on classification errors. I construct sharp, bias-aware confidence intervals, easy to compute and uniformly valid over misclassification patterns compatible with bounds on group-wise misclassification rates. Under some conditions, combining two classifiers provides valid standard inference without knowledge of the misclassification pattern.