The contents of this dissertation are split into three chapters, each of which covers a distinct problem in econometrics.
The first chapter considers the problem of hypothesis testing in a weakly identified instrumental variables models with a potentially large number of instrumental variables. Instrumental variables strategies, where a causal effect is identified by exploiting exogenous variation in the explanatory variable induced by changes in instrumental variables, are one of the most common quasi-experimental research designs used in economics. In recent years, there has been interest in using a large number of instruments in combination with some regularized method, such as LASSO, in order to flexibly model the relationship between the instrumental and explanatory variable. However, in these setting there has been little work on testing hypotheses about the structural parameter when this first-stage relationship is weak. In this chapter I propose a new test for the structural parameter in a instrumental variables models that has correct asymptotic sizeeven when the number of instruments is potentially much larger than the sample size and identification is arbitrarily weak. The limiting distribution of the test statistic is derived through a novel direct Gaussian approximation argument and is combined with the sup-score test in order to improve power against certain alternatives. In both empirical data and simulation study the proposed methods are shown to have favorable size control and power properties compared to existing methods.
In the second chapter, coauthored with Adam Baybutt, we consider inference on the conditional average treatment effect (CATE) under first stage model misspecification. Plausible identification of CATEs can rely on controlling for a large number of variables to account for confounding factors. In these high-dimensional settings, estimation of the CATE requires estimating first-stage models whose consistency relies on correctly specifying their parametric forms. While doubly-robust estimators of the CATE exist, inference procedures based on the second stage CATE estimator are not doubly robust. Using the popular augmented inverse propensity weighting signal, we propose an estimator for the CATE whose resulting Wald-type confidence intervals are doubly robust. We assume a logistic model for the propensity score and a linear model for the outcome regression, and estimate the parameters of these models using a Lasso penalty to address the high dimensional covariates. Our proposed estimator remains consistent at the nonparametric rate and our proposed pointwise and uniform confidence intervals remain asymptotically valid even if one of the logistic propensity score or linear outcome regression models are misspecified.
The final chapter, coauthored with Prof. Rodrigo Pinto, investigates the relationship among monotonicity conditions in IV models with multiple choices and categorical instruments. The comparison between monotonicity conditions of ordered and unordered choice models is central to our analysis. We show that these seemingly unrelated conditions exhibit non-trivial symmetries that can be traced back to a weaker condition called Minimal Monotonicity. This novel condition captures an essential property for identifying causal parameters while being necessary for ascribing causal interpretation to Two-Stage Least Squares (2SLS) estimands. We show that minimal monotonicity naturally arises from a notion of rationality in revealed preference analysis. The condition enables to describe non-standard choice behaviors and serves as a building block for a wide range of economically-justified monotonicity conditions that do not fit the narrative dictated by either ordered or unordered choice models.
