arXiv 9 Apr 2026 · Statistics — Methodology
arXiv:2604.08798 · PDF · DOI · OpenAlex · Extracted main text
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 $τ$ 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 $τ$. 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 $δ$, the bias is bounded by $|τ|\,\mathbb{E}[|2p-1|]\,δ\,(2V^*)^{-1}$, 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.
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| Reference | Intensity | Mentions | Sections | Main text | |
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| 1 | Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C… (2018) Double/debiased machine learning for treatment and structural parameters | 0.843 | 4 | 4 | 75% |
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| 9 | Robinson, P.M (1988) Root-$N$-consistent semiparametric regression | 0.405 | 1 | 1 | 100% |
| 10 | Schennach, S.M (2016) Recent advances in the measurement error literature | 0.405 | 1 | 1 | 100% |
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