Inference methods for unit-specific coefficients in panel data models with latent group structure

This paper introduces statistical inference procedures for unit-specific coefficients in panel data models, where the coefficients exhibit a latent group structure. The proposed methods achieve efficiency gains by clustering units into a small number of groups, while explicitly accounting for the statistical uncertainty of group assignments. The core idea is to integrate standard inference procedures, such as the $t$-test and Wald tests, with confidence sets for group membership. Two methods are proposed: the first takes the minimum of the test statistics over the confidence set for group membership, and the second corrects for bias caused by possible group misassignment. The former can produce shorter but possibly disconnected sets, while the latter guarantees connected, interpretable intervals at some cost in length. We also develop standard errors that are adjusted for possible group misassignment and valid even with short time periods, which may be of independent interest. Monte Carlo simulations demonstrate that our approach yields narrower confidence sets for units with relatively large error variances than unit-by-unit time-series methods. In contrast, ignoring statistical uncertainty in the group membership estimation leads to distortions in size and coverage. We illustrate the method with an empirical example that estimates the effect of the minimum wage in each U.S. state.