This dissertation contains three essays in econometrics. A common theme is robust estimation and inference in non-standard settings, with a particular focus on difference-in-differences and related research designs.The first chapter investigates the common practice of testing for pre-existing trends ("pre-trends") in difference-in-differences and related research designs. These tests are commonly used to assess the plausibility of the parallel trends assumption needed for causal identification. I show both theoretically and in simulations based on a systematic review of recent papers that pre-trends testing has two important limitations. First, the power of pre-tests against relevant violations of parallel trends may be low. Second, the distribution of conventional estimates conditional on passing the pre-test differs from its unconditional distribution. In non-pathological cases, the bias and coverage rates of conventional estimates and confidence intervals are worse conditional on passing the pre-test. I discuss alternative approaches to estimation and inference in settings where there is concern that parallel trends may be violated. I also introduce new methodology for retrospectively analyzing research that has been screened on the basis of pre-trends.The second chapter, which is joint work with Ashesh Rambachan, introduces new methodology for causal inference in settings where there is concern that the parallel trends assumption may be violated. We introduce restrictions on the possible violations of parallel trends that formalize the intuition behind the common practice of testing for pre-trends discussed in Chapter 1. These restrictions formalize the idea that the pre-treatment differences in trends are informative about what would have happened under the counterfactual that treatment had not occurred. Under these restrictions, the causal parameter of interest is typically set-identified. We introduce new methodologies for inference (based in part on the work in Chapter 3) that we show have good theoretical properties -- including optimal local asymptotic power for many parameter configurations -- and perform well in simulations based on recent papers. We recommend that researchers use our methods to conduct sensitivity analyses that make precise what needs to be assumed about the possible differences in trends to draw a given conclusion.Finally, Chapter 3 -- which is joint work with Isaiah Andrews and Ariel Pakes -- provides general methodology for inference using conditional moment inequalities with nuisance parameters that enter linearly. This structure arises in the context of robust inference in difference-in-differences settings discussed in Chapter 2, as well as in a variety of settings in industrial organization. A key advantage of our approach is that it allows for fast computation even in settings where the dimension of the nuisance parameter is large.
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