Fragility of Joint Identification in the Roy Model : A Note on Deterministic Sorting and Stochastic Selection

Heckman and Honoré (1990) identify the joint distribution of latent sector-specific outcomes in a nonparametric Roy model. This note isolates the measurement condition behind that result. The identifying force is not the exclusion restriction alone, but the exclusion restriction combined with the known deterministic observation rule D = 1{Y₠> Y₀}, which reveals the side of a moving boundary in the latent plane. The main result shows that this rule is a singular measurement condition. Under a local η-relaxation of deterministic Roy sorting, an unknown stochastic selection kernel opens an absorption face of observationally equivalent latent joint laws whose total-variation diameter is linear in η, and this linear rate is exact on the face. A point-to-set corollary formalizes the boundary: at η = 0 the Heckman–Honoré restrictions deliver a singleton joint law, whereas every total-variation neighborhood of the exact Roy observed law contains stochastic-Roy laws whose latent identified sets have first-order diameter. The note also records the sharp global geometry that remains after deterministic sorting is relaxed: the latent-law identified set is a submeasure linear program formed from treated and untreated joint submeasures. This global characterization is supporting geometry for the fragility result, not the main message. The results distinguish deterministic Roy tomography from ordinary partial identification u nder stochastic selection.