Regression discontinuity designs assess causal effects in settings where treatment is determined by whether an observed running variable crosses a pre-specified threshold. Here we propose a new approach to identification, estimation, and inference in regression discontinuity designs that uses knowledge about exogenous noise (e.g., measurement error) in the running variable. In our strategy, we weight treated and control units to balance a latent variable of which the running variable is a noisy measure. Our approach is explicitly randomization-based and complements standard formal analyses that appeal to continuity arguments while ignoring the stochastic nature of the assignment mechanism.

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