Identification of Latent Group Effects under Conditional Calibration

TL;DR

This paper develops a method to identify a latent group effect using observed calibrated probabilities, providing explicit formulas, conditions for identification, and analyzing bias under calibration errors.

Contribution

It introduces a simple ratio of weighted moments for identifying the latent group coefficient and characterizes when identification fails or differs from marginal effects.

Findings

The latent-group coefficient is point-identified via a ratio of weighted moments.

Identification fails if the score is deterministic in covariates.

Bias under calibration error is bounded and depends on calibration error magnitude.

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 $p\in[0,1]$ satisfying $\mathbb{E}[G|p,X]=p$. 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 $|\tau|\,\mathbb{E}[|2p-1|]\,\delta\,(2V^*)^{-1}$, a bound that is sharp over all calibration error functions of that magnitude. Hard-threshold classification at $p=1/2$ attenuates the estimated gap by a factor strictly less than one. Monte Carlo experiments confirm the asymptotic theory, trace the divergence of RMSE as $V^*\to 0$, illustrate the attenuation bias of hard-threshold classification, and verify identification of the variance-weighted estimand under heterogeneous effects.