Identification of Latent Group Effects under Conditional Calibration

Abstract

We study identification of a structural group effect when the group indicator $G\in \{0,1\}$ is unobserved, but the analyst observes a calibrated probability score $p$ satisfying $E[G\mid p,X]=p$. Under a constant-coefficient structural mean model, the latent-group coefficient $\ddot{I}„$ is point-identified by a closed-form ratio of observable moments whose denominator is the residual score variance $V^{\ast }=E[(p-E[p\mid X]{)}^2]$. Identification fails exactly when the score is a deterministic function of $X$; we construct an explicit continuum of observationally equivalent models showing the failure is genuine. The marginal latent mean gap decomposes as $\ddot{I}„$ plus a compositional term that is itself identified in closed form, and we characterise when the two coincide. The oracle estimator is $\sqrt{n}$-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded by $\hat{I}´$, the bias obeys a sharp bound proportional to $\hat{I}´/V^{\ast }$, and hard-threshold classification attenuates the estimated gap. Monte Carlo experiments confirm the theory, including the variance-weighted estimand under heterogeneous effects.

41 pages, 5 figures, 5 tables