Inference After Model Refutation: Uniform Post-Relaxation Inference for Partially Identified Models

Abstract

When data reject an econometric model, the researcher must decide which identifying assumptions to give up. If several equally preferred repairs restore compatibility, the repair itself is unknown, and inference under any one of them can exclude structural values supported by the others, even when that inference is exact. I propose Post-Relaxation Projection (PR), which carries one simultaneous confidence event through every repair that may be minimal. With repair-by-repair primitive bounds, PR equals the union of all post-relaxation targets compatible with the band, so no smaller band-based set has the same guarantee. I derive limits on learning the repair, the exact perturbation radius of the minimal-repair correspondence, oracle-equivalent adaptation, and a reduced-form version valid at zero first stages. In cross-country data on institutions and income, the data do not settle which historical instrument should be dropped, yet every admissible minimum repair implies a positive effect of institutions.