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  1. Summary

    Modern empirical work in regression discontinuity (RD) designs often employs local polynomial estimation and inference with a mean square error (MSE) optimal bandwidth choice. This bandwidth yields an MSE-optimal RD treatment effect estimator, but is by construction invalid for inference. Robust bias-corrected (RBC) inference methods are valid when using the MSE-optimal bandwidth, but we show that they yield suboptimal confidence intervals in terms of coverage error. We establish valid coverage error expansions for RBC confidence interval estimators and use these results to propose new inference-optimal bandwidth choices for forming these intervals. We find that the standard MSE-optimal bandwidth for the RD point estimator is too large when the goal is to construct RBC confidence intervals with the smaller coverage error rate. We further optimize the constant terms behind the coverage error to derive new optimal choices for the auxiliary bandwidth required for RBC inference. Our expansions also establish that RBC inference yields higher-order refinements (relative to traditional undersmoothing) in the context of RD designs. Our main results cover sharp and sharp kink RD designs under conditional heteroskedasticity, and we discuss extensions to fuzzy and other RD designs, clustered sampling, and pre-intervention covariates adjustments. The theoretical findings are illustrated with a Monte Carlo experiment and an empirical application, and the main methodological results are available in R and Stata packages.

     
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  2. The regression discontinuity (RD) design is one of the most widely used nonexperimental methods for causal inference and program evaluation. Over the last two decades, statistical and econometric methods for RD analysis have expanded and matured, and there is now a large number of methodological results for RD identification, estimation, inference, and validation. We offer a curated review of this methodological literature organized around the two most popular frameworks for the analysis and interpretation of RD designs: the continuity framework and the local randomization framework. For each framework, we discuss three main topics: ( a) designs and parameters, focusing on different types of RD settings and treatment effects of interest; ( b) estimation and inference, presenting the most popular methods based on local polynomial regression and methods for the analysis of experiments, as well as refinements, extensions, and alternatives; and ( c) validation and falsification, summarizing an array of mostly empirical approaches to support the validity of RD designs in practice. 
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  3. In this article, we introduce the Stata (and R) package rdmulti, which consists of three commands (rdmc, rdmcplot, rdms) for analyzing regression-discontinuity (RD) designs with multiple cutoffs or multiple scores. The command rdmc applies to noncumulative and cumulative multicutoff RD settings. It calculates pooled and cutoff-specific RD treatment effects and provides robust biascorrected inference procedures. Postestimation and inference is allowed. The command rdmcplot offers RD plots for multicutoff settings. Finally, the command rdms concerns multiscore settings, covering in particular cumulative cutoffs and two running variable contexts. It also calculates pooled and cutoff-specific RD treatment effects, provides robust bias-corrected inference procedures, and allows for postestimation and inference. These commands use the Stata (and R) package rdrobust for plotting, estimation, and inference. Companion R functions with the same syntax and capabilities are provided.

     
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