Identification of Latent Group Effects under Conditional Calibration

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 $p\in [0,1]$ satisfying $\mathbb{E}[G\mathrm{\mid }p,X]=p$. Under a constant-coefficient structural mean model, the latent-group coefficient $\tau$ is point-identified from the joint law of observables $(Y,X,p)$ by a simple ratio of weighted moments: the covariance of the signed score $2p-1$ with the covariate-partialled outcome, divided by twice the residual variance of the score after conditioning on covariates. Identification fails if and only if the score is a deterministic function of $X$; we establish this by constructing an explicit continuum of observationally equivalent models indexed by arbitrary values of $\tau$. The identified coefficient differs from the marginal latent mean gap by a compositional term that is unidentified without further assumptions; we give a necessary and sufficient condition for the two to coincide. The oracle estimator is $\sqrt{n}$-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded uniformly by $\delta$, the bias is bounded by $\mathrm{\mid }\tau \mathrm{\mid }\mathbb{E}[\mathrm{\mid }2p-1\mathrm{\mid }]\delta (2V^{\ast }{)}^{-1}$, a bound that is sharp over all calibration error functions of that magnitude. Hard-threshold classification at $p=1\mathrm{/}2$ attenuates the estimated gap by a factor strictly less than one. Monte Carlo experiments confirm the asymptotic theory, trace the divergence of RMSE as $V^{\ast }\to 0$, illustrate the attenuation bias of hard-threshold classification, and verify identification of the variance-weighted estimand under heterogeneous effects.