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This dissertation consists of three essays. The first essay focuses on regression discontinuity with a donut, the second essay looks at spillovers in synthetic controls, and the third essay examines a new two-sample test. Regression discontinuity (RD) designs use policy thresholds to identify the causal effects of policy. RD Donut designs allow identification in situations with some manipulation, but they require extrapolation, typically projecting a polynomial, to identify treatment effects. Chapter 1 extrapolates into a donut by leveraging high-level smoothness conditions similar to those used to find optimal bandwidths. I start using known derivative bounds before using data-determined bounds. The synthetic control method (SCM) is often used in treatment effect estimation with panel data where only a few units are treated and a small number of post-treatment periods are available. Current estimation and inference procedures for SCM do not allow for the existence of spillover effects. In a related paper [Cao and Dowd, 2021], we consider estimation and inference for SCM, allowing for spillover effects. In Chapter 2 we show simulations and empirical examples of this method. Empirical cumulative distribution functions are used to test the hypothesis that two samples come from the same distribution. Chapter 3 describes a statistic that is usable under the same conditions as the Kolmogorov-Smirnov test, but provides more power than other extant tests in that vein. I prove the validity of the procedure, provide code, and show several simulations demonstrating substantial power.

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