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 $p$ satisfying $E[G\mid p,X]=p$.

Under a constant-coefficient structural mean model, the latent-group coefficient $\tau$ is point-identified by a closed-form ratio of observable moments whose denominator is the residual score variance $V^{*}=E[(p-E[p\mid X])^2]$.

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 $\tau$ 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 $\delta$, the bias obeys a sharp bound proportional to $\delta/V^{*}$, and hard-threshold classification attenuates the estimated gap.

Monte Carlo experiments confirm the theory, including the variance-weighted estimand under heterogeneous effects.