Extension to continuous outcomes as the discretization radius vanishes

Establish identification and valid inference for the average treatment effect when the target outcome is continuous by extending the ε-cover approach to the limit ε → 0, thereby removing discretization of the outcome space. Specifically, derive conditions under which the bias bound |θ_w − \tilde{θ}_w(ε)| vanishes as ε → 0 and construct confidence intervals that remain valid in this limit, extending Theorem prop:continuous which currently fixes ε > 0 to ensure a finite cover.

Background

The paper develops identification, estimation, and inference for treatment effects when outcomes are not observed in the experimental sample and must be inferred using remotely sensed variables. For continuous outcomes, the authors construct an ε-cover of the outcome support and provide identification and inference results that are valid for fixed ε > 0, along with a worst-case bias bound for the average effect.

To ensure finite-dimensional representations and tractable inference, Theorem prop:continuous fixes ε > 0. The authors explicitly note that extending the analysis to let ε → 0 is not addressed and is left for future research, which would remove discretization and potentially yield sharper results for continuous outcomes.