Regression Discontinuity Designs with Complex Data

In modern medical practice, it is common to regularly monitor patients' fasting blood sugar, and declare them to have prediabetes (and encourage lifestyle changes) if this number crosses a pre-specified threshold. The sharp, threshold-based treatment policy suggests that we should be able to estimate the long-term benefit of care given to prediabetic patients by comparing health trajectories of patients with blood-sugar measurements right above and below the threshold. A naive regression-discontinuity analysis, however, is not applicable here, as it ignores the temporal dynamics of the problem where, e.g., a patient just below the threshold on one visit may become prediabetic (and receive treatment) following their next visit.

Here, we study dynamic thresholding designs in Markovian systems, and show that a regression-discontinuity estimator run on aggregate discounted outcomes can still be used to identify a relevant causal target, namely the policy gradient of moving the treatment threshold. We develop results for estimation and inference of this target, and discuss implications of our findings to interpretation of regression-discontinuity studies in preventive healthcare. More broadly, our results highlight the promise of adapting widely used observational study techniques to dynamic systems. 

Boundary Discontinuity Designs are used to learn about treatment effects along a continuous boundary that splits units into control and treatment groups. This design corresponds to a Multi-Score Regression Discontinuity Design, a leading special case being the Geographic Regression Discontinuity Design. We study the statistical properties of local polynomial treatment effects estimators along the continuous treatment assignment boundary. We consider two distinct local polynomial methods: one based explicitly on the bivariate location variable of units relative to the treatment assignment boundary, and the other based on their univariate distance to the boundary. For each approach, we develop pointwise and uniform estimation and inference methods for the treatment effect function over the assignment boundary. We show that methods based on distance can have substantially worse misspecification biases when the boundary exhibits kinks or other irregularities. Our methods deliver valid heat maps with feasible confidence bands for treatment effects over the boundary, as well as feasible hypothesis tests of the shape of the treatment effect curve (e.g., testing for monotonic treatment effects along the boundary). We illustrate our methods with an empirical application and in simulations. 

Clustered sampling is prevalent in empirical regression discontinuity (RD) designs, but it did not gain much attention in the theoretical literature. In this paper, we introduce a general model-based framework for such settings and derive high-level conditions under which the standard local linear RD estimator is asymptotically normal. We show that clustered standard errors that are currently used in practice can be overly conservative in finite samples. To address these issues, we propose a new nearest-neighbor-type variance estimator. We verify that our high-level assumptions hold across a wide range of empirical designs, including settings of growing cluster sizes, and demonstrate the estimator's finite-sample performance via simulations and a diverse set of applications. 

This paper studies identification and estimation of regression discontinuity (RD) extrapolation processes that measure policy effects away from the running variable cutoff. Our proposed semi-parametric identification strategy uses weaker assumptions than those previously adopted in the literature and, at the same time, enjoys a new robustness property of reducing to classic nonparametric RD identification when the magnitude of extrapolation goes to zero. For estimation and inference, we propose a doubly-robust two-step procedure that provides first-step bias-correction as well as valid inference. We applied our proposed method to extend the empirical analysis in Lindo, Sanders, and Oreopoulos (2010) on college academic probation. Our method allows us to estimate the effect of academic probation for students not exactly at the probation GPA cutoff.