Design-Based Inference under Random Potential Outcomes via Riesz Representation

We introduce a design-based framework for causal inference that accommodates random potential outcomes, thereby extending the classical Neyman-Rubin model in which outcomes are treated as fixed. Each unit's potential outcome is modelled as a structural mapping $\tilde{y}_i(z, \omega)$, where $z$ denotes the treatment assignment and \(\omega\) represents latent outcome-level randomness. Inspired by recent connections between design-based inference and the Riesz representation theorem, we embed potential outcomes in a Hilbert space and define treatment effects as linear functionals, yielding estimators constructed via their Riesz representers. This approach preserves the core identification logic of randomised assignment while enabling valid inference under stochastic outcome variation. We establish large-sample properties under local dependence and develop consistent variance estimators that remain valid under weaker structural assumptions, including partially known dependence. A simulation study illustrates the robustness and finite-sample behaviour of the estimators. Overall, the framework unifies design-based reasoning with stochastic outcome modelling, broadening the scope of causal inference in complex experimental settings.