Sharp Identification for Regression with Interval-Observed and Missing Covariates

Researchers frequently estimate regressions when key variables are missing or observed only through intervals. Imputation or assumptions about the observation process can yield seemingly precise but misleading conclusions. We develop methods for identification and inference that remain robust to such incomplete data. For a broad class of conditional moment models, we characterize the sharp identified set, the set of all parameter values consistent with the observed data and maintained regression model, without restricting how incomplete observations arise. We establish Hausdorff-consistent estimation and uniformly valid confidence regions, and develop computationally tractable implementations for common regression models. In an application relating interval-elicited discount rates to on-time high-school graduation, conventional inference excludes zero, whereas our interval-data-robust confidence regions include it. Exploiting sharp identification nevertheless yields substantially tighter confidence regions than non-sharp alternatives.