A Design-Based Riesz Representation Framework for Randomized Experiments

Abstract

We describe a new design-based framework for drawing causal inference in randomized experiments. Estimands in the framework are defined as arbitrary linear functionals of the potential outcome functions, which are posited to live in an experimenter-specified function class. This makes the framework expressive, allowing experimenters to formulate and investigate a wide range of causal questions. We describe a class of estimators for estimands defined using the framework and investigate their properties. The construction of the estimators is based on insights from the Riesz representation theorem. We provide necessary and sufficient conditions for unbiasedness and consistency. Finally, we provide conditions under which the estimators are asymptotically normal, and describe a conservative variance estimator to facilitate inference about the estimands.