This dissertation consists of three chapters that study treatment effect estimation and treatment choice learning under the potential outcome framework (Neyman, 1923; Rubin, 1974). The first two chapters study how to efficiently combine an experimental sample with an auxiliary observational sample when estimating treatment effects. In chapter 1, I derive a new semiparametric efficiency bound under the two-sample setup for estimating ATE and other functions of the average potential outcomes. The efficiency bound for estimating ATE with an experimental sample alone is derived in Hahn (1998) and has since become an important reference point for studies that aim at improving the ATE estimation. This chapter answers how an auxiliary sample containing only observable characteristics (covariates, or features) can lower this efficiency bound. The newly obtained bound has an intuitive expression and shows that the (maximum possible) amount of variance reduction depends positively on two factors: 1) the size of the auxiliary sample, and 2) how well the covariates predict the individual treatment effect. The latter naturally motivates having high dimensional covariates and the adoption of modern machine learning methods to avoid over-fitting.

In chapter 2, under the same setup, I propose a two-stage machine learning (ML) imputation estimator that achieves the efficiency bound derived in chapter 1, so that no other regular estimators for ATE can have lower asymptotic variance in the same setting. This estimator involves two steps. In the first step, conditional average potential outcome functions are estimated nonparametrically via ML, which are then used to impute the unobserved potential outcomes for every unit in both samples. In the second step, the imputed potential outcomes are aggregated together in a robust way to produce the final estimate. Adopting the cross-fitting technique proposed in Chernozhukov et al. (2018), our two-step estimator can use a wide range of supervised ML tools in its first step, while maintaining valid inference to construct confidence intervals and perform hypothesis tests. In fact, any method that estimates the relevant conditional mean functions consistently in square norm, with no rate requirement, will lead to efficiency through the proposed two-step procedure. I also show that cross-fitting is not necessary when the first step is implemented via LASSO or post-LASSO. Furthermore, our estimator is robust in the sense that it remains consistent and root n normal (no longer efficient) even if the first step estimators are inconsistent.

Chapter 3 (coauthored with Kirill Ponomarev) studies model selection in treatment choice learning. When treatment effects are heterogeneous, a decision maker, given either experiment or quasi-experiment data, can attempt to find a policy function that maps observable characteristics to treatment choices, aiming at maximizing utilitarian welfare. When doing so, one often has to pick a constrained class of functions as candidates for the policy function. The choice of this function class poses a model selection problem. Following Mbakop and Tabord-Meehan (2021) we propose a policy learning algorithm that incorporates data-driven model selection. Our method also leverages doubly robust estimation (Athey and Wager, 2021) so that it could retain the optimal root n rate in expected regret in general setups including quasi-experiments where propensity scores are unknown. We also refined some related results in the literature and derived a new finite sample lower bound on expected regret to show that the root n rate is indeed optimal.

This paper focuses on nonseparable structural models of the form Y = m(X, U, α0) with U X and in which the structural parameter α0 contains both finite dimensional (θ0) and infinite dimensional (h0) unknown components. Our proposal is to estimate α0 by a minimum distance from independence (MDI) criterion. We show that: (i) our estimator of h0 is consistent and obtain rates of convergence; (ii) the estimator of θ0 is square root n consistent and asymptotically normally distributed.

This dissertation studies a few econometric theories potentially useful for applied economists. In the first two chapters, I study estimation and inference in a semi-parametric model under a monotonicity restriction on the non-parametric component. I develop a new semi-parametric estimator that can be implemented without choosing any smoothing parameters and construct a confidence band for the non-parametric component under monotonicity. The finite dimensional parametric estimator satisfies asymptotic normality. The asymptotic distribution of $L_{\infty}$ - distance for the non-parametric component is Gumbel with the rate of convergence $O_p((\frac{\log n}{n})^{1/3})$. I apply the estimator to estimate the returns to schooling under the restriction that age has monotonic effect on wages. The confidence interval of the returns to schooling and the confidence band of the age effect on the log of wage under an assumed monotonic relationship are reported. I illustrate the confidence intervals of the semi-parametric estimator and the confidence band of the semi-nonparametric estimator using Monte Carlo simulations.\On the last chapter, my coauthors and I propose a pragmatic approach to the errors-in-variables and nonlinear panel models. These models are often deemed impossible to estimate in their most general forms. For example, the higher order moments approach to errors-in-variables model fails when there is conditional heteroscedasticity. Similarly, nonlinear panel models with fixed effects and small T are known to be problematic to estimate. We propose estimating these models using approximate moments, using a Taylor series approximation applied to Kadane's (1971) small sigma approach. Simulation results suggest that the approximation leads to reasonable sampling properties. Our proposal complements the newly resurgent literature on sensitivity analysis.

The chapters of this dissertation are devoted to three different topics. The first chapter studies estimation of parameters expressed via non-differentiable functions. Such parameters are abundant in econometric models and typically take the form of maxima or minima of some estimable objects. Examples include bounds on the average treatment effects in non-experimental settings, identified sets for the coefficients in regression models with interval-valued data, bounds on the distribution of wages accounting for selection into employment, and many others. I consider estimators of the form $\phi(\hat{\theta}_n + \hat{v}_{1, n}) + \hat{v}_{2, n}$, where $\hat{\theta}_n$ is the efficient estimator for $\theta_0$, and $\hat{v}_{1, n}, \hat{v}_{2, n}$ are suitable adjustment terms. I characterize the optimal adjustment terms and develop a general procedure to compute them from the data. A simulation study shows that the proposed estimator can have lower finite-sample bias and variance than the existing alternatives. As an application, I consider estimating the bounds on the distribution of valuations and the optimal reserve price in English auctions with independent private values. Empirically calibrated simulations show that the resulting estimates are substantially sharper than the previously available ones. The second chapter studies inequality selection in partially identified models. Many partially identified models have the following structure: given a parameter vector and covariates, the model produces a set of predictions while the researcher observes a single outcome. Examples include entry games with multiple equilibria, network formation models, discrete-choice models with endogenous explanatory variables or heterogeneous choice sets, and auctions. Sharp identified sets for structural parameters in such models can be characterized via a special kind of moment inequalities. For a given parameter value, the inequalities verify that the observed conditional distribution of the outcome given covariates belongs to the set of distributions admitted by the model. In practice, checking all of the inequalities is often computationally infeasible, and many of them may not even be informative. Therefore, some inequality selection is required. In this chapter, I propose a new analytical criterion that dramatically reduces the number of inequalities required to characterize the sharp identified set. In settings where the outcome space is finite, I characterize the smallest subset of inequalities that guarantees sharpness and show that it can be efficiently computed using graph propagation techniques. I apply the proposed criterion in the context of market entry games, network formation, auctions, and discrete-choice. The third chapter (coauthored with Liqiang Shi) is about model selection for policy learning. When treatment effects are heterogeneous, a decision-maker that has access to (quasi-)experimental data can attempt to find the optimal policy function, mapping observable characteristics into treatment choices, to maximize utilitarian welfare. When several different policy classes are available, the choice of the policy class poses a model selection problem. In this chapter, following Athey and Wager (2021) and Mbakop and Tabord-Meehan (2021), we propose a policy learning algorithm that leverages doubly-robust estimation and incorporates data-driven model selection. We show that the proposed algorithm automatically selects the best available class of policies and achieves the optimal $n^{-1/2}$ rate of convergence in terms of expected regret. We also refine some of the existing related results and derive a new finite-sample lower bound on expected regret.

This dissertation contains two chapters. The first chapter studies causal parameters that depend on a moment of the joint distribution of potential outcomes. Such parameters are especially relevant in policy evaluation settings, where noncompliance is common and accommodated through the model of Imbens & Angrist (1994). The sharp identified set for these parameters is an interval with endpoints characterized by the value of optimal transport problems. Sample analogue estimators are proposed based on the dual problem of optimal transport. These estimators are root-n consistent and converge in distribution under mild assumptions. Inference procedures based on the bootstrap are straightforward and computationally convenient. The ideas and estimators are demonstrated in an application revisiting the National Supported Work Demonstration job training program. Estimates suggest that workers who would see below average earnings without treatment tend to see above average benefits from treatment.

The second chapter proposes a methodology for studying the robustness of results drawn from incomplete datasets. Selection is measured as the squared Hellinger divergence between the distributions of complete and incomplete observations, which has a natural interpretation. The breakdown point is defined as the minimal amount of selection needed to overturn a given result. Reporting point estimates and lower confidence intervals of the breakdown point is a simple, concise way to communicate a result’s robustness. An estimator of the breakdown point of results drawn from GMM models is proposed and shown root-n consistent and asymptotically normal under mild assumptions. Lower confidence intervals of the breakdown point are simple to construct. The chapter concludes with a simulation study illustrating the good finite sample performance of the procedure.

This dissertation consists of three chapters that explore new methodologies and applications in econometrics.

In the first chapter, I propose a new class of functions defined by the so-called approximate sparsity condition. In general, functions in well-known classes can often be characterized by the rate of decay of their Fourier coefficients. The approximate sparsity condition generalizes this characterization by considering all sequences of such coefficients decreasing to zero at a certain rate while allowing for reordering. In particular, this generalization can potentially accommodate the modeling uncertainty of the unknown functions and aid estimation. For this new class of functions, I establish the metric entropy and minimax rate of convergence in terms of the estimation error. Moreover, I propose a data-driven density estimator based on a thresholding procedure and show this estimator can achieve the minimax rate up to a log term. A simulation study is also provided to demonstrate the performance of this estimator.

The second chapter focuses on the crucial role of conditional density in economic applications and introduces a data-driven nonparametric conditional density estimator suitable for high-dimensional covariates. I first demonstrate that conditional density can be represented as a series, with each series term consisting of a known function multiplied by its conditional expectation. This structure is particularly beneficial in high-dimensional settings, where these conditional expectations can be flexibly estimated using various machine learning methods. Subsequently, I detail an algorithm that outlines the construction of my estimator based on this series formulation. Specifically, this procedure involves estimating a large number of conditional expectations and selecting the series cutoff through a data-driven procedure based on cross-validation. Lastly, I establish a general theory showing that this data-driven estimator is asymptotically optimal and can accommodate a wide range of machine learners under mild assumptions.

In the third chapter, I extend difference-in-differences to settings involving continuous treatments. Specifically, I identify the average treatment effect on the treated (ATT) at any level of continuous treatment intensity, using a conditional parallel trends assumption. In this framework, estimating the ATTs requires first estimating infinite-dimensional nuisance parameters, such as the conditional density of the continuous treatment, which can introduce significant biases. To address this challenge, I propose estimators for the causal parameters under the double/debiased machine learning framework. I demonstrate that these estimators are asymptotically normal and provide consistent variance estimators. To illustrate the effectiveness of my methods, I reexamine the study by Acemoglu and Finkelstein (2008), which assessed the effects of the 1983 Medicare Prospective Payment System (PPS) reform. By reinterpreting their research design using a difference-in-differences approach with continuous treatment, I nonparametrically estimate the treatment effects of the 1983 PPS reform, thereby providing a more detailed understanding of its impact.

This dissertation studied two main topics: inference methods for directionally differentiable functions with first order degeneracy and estimation methods for semiparametric instrumental variables (IV) models with partial identication. Chapter 1 presents a general asymptotic framework for conducting inference on directionally differentiable functions with a zero first order derivative. Chapter 2 applies the theory to develop improved inference methods for the rank of a matrix, which is important in estimation and inference IV models.

Chapter 3 studies robust and optimal estimation in a canonical semiparametric IV model --- partially linear IV model --- with nonparametric partial identification.

The first two chapters of this dissertation study identification of a new class of demand models termed \textit{perturbed utility models}. The first chapter provides sufficient conditions under which structural functions in these models can be uniquely determined from knowledge of conditional means. The second chapter proposes a definition of complementarity/substitutability for these models and shows how to recover this measure from data.

The third chapter of this dissertation studies inference in a class of partially identified models. Specifically, this chapter provides a finite-sample power comparison between two existing tests of moment inequalities.