This paper develops an auxiliary-measurement approach to identifying average treatment effects in generalized Roy environments where treatment choice may depend directly on potential outcomes. Identification is formulated as a primal–dual inverse problem. A latent selection-odds representer anchors a causally correct element in an observed calibration set, which may be nonunique. An adjoint outcome representer certifies that the target mean is invariant over that set, so identification does not require point identification of the calibrating function itself. This separation yields a trichotomy between non-invariance, irregular identification, and regular orthogonal-moment representation, according to the position of the outcome signal in the adjoint range. The same geometry delivers an orthogonal estimating equation and an exact product-bias identity, supporting sieve GMM and cross-fitted estimation. Simulations illustrate regular, weak, and failed range regimes.
An application to retirement and cognition in the Health and Retirement Study shows that specifications restricted to observed adjustment and those allowing selection on gains yield materially different estimates, illustrating the framework’s empirical content under maintained calibration and adjoint-representation assumptions.