Exact inference after data-driven control-unit selection in difference-in-differences

Abstract

In comparative case studies that select a single control using a centered pre-treatment fit criterion, researchers report a difference-in-differences interval treating the selection as fixed. Under a Gaussian panel model with known covariance, a centered criterion yields exact naive intervals when the covariance is separable with a spherical time factor, for any cross-sectional covariance. Otherwise, given the selection and an independent residual, the estimator is truncated normal; the pivot yields exact selective coverage. In our simulations, naive coverage was often conservative and never materially below nominal; working-model plug-in inference exhibited modest undercoverage under strong serial correlation.