UCLA

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.