This dissertation investigates semiparametric social interaction models. The goal is to identify and estimate the endogenous social interactions using a flexible semiparametric model to control for confounding factors. The rationale for considering nonparametric controls is that, if the groups or networks are not randomly assigned, or if the contextual effects are heterogeneous, identifying the endogenous social interaction effect is difficult without adequate controls. This thesis contains two chapters.

Chapter 1 first studies the identification of the endogenous social interaction effect in the semiparametric models. The identification is attained by using the instrumental variable (IV) approach after partialling out the nonparametric controls. To estimate the endogenous social interaction effect, I propose a semiparametric two-step generalized method of moments (GMM) estimator with the optimal weight matrix clustered at the group or network level. This chapter focuses on the semiparametric estimators that use the first step series method. The primitive regularity conditions are provided for the consistency and asymptotic normality of the semiparametric series GMM estimators.

In Chapter 2, I apply more flexible machine learning methods in the first step nonparametric estimation to detect severe nonlinearities and higher-order interactions, including LASSO, Random Forest, and Neural Nets. %The asymptotic properties of the semiparametric machine learning estimators are proved given high level conditions. Monte Carlo simulations are conducted to investigate the finite sample performance of semiparametric estimators using different first-step Machine Learning methods. The results suggest that no estimation method dominates across all the Data Generating Processes (DGPs) considered. It is also reflected in the simulation results that the debiased estimators using first step post-LASSO or Neural Nets methods are more reliable and performs relatively well across the settings considered. For this reason, these two debiased estimators are recommended for use in empirical studies.

This dissertation studies econometric questions in the context of three different methods that are frequently used by empirical economists.

Chapter 1 provides a short introduction to the contexts, questions, methods and results studied in Chapter 2 to Chapter 4.

Chapter 2 studies a nonparametric hedonic equilibrium model in which certain product characteristics are unobserved. Unlike most previously studied hedonic models, both the observed and unobserved agent heterogeneities enter the structural functions nonparametrically. Prices are endogenously determined in equilibrium. Using both within-market and cross-market price variation, I show that all the structural functions of the model are nonparametrically identified up to normalization. In particular, the unobserved product quality function is identified if the relative prices of the agent characteristics differ in at least two markets. Following the constructive identification strategy, I provide easy-to-implement series minimum distance estimators of the structural functions and derive their uniform rates of convergence. To illustrate the estimation procedure, I estimate the unobserved efficiency of American full-time workers as a function of age and unobserved ability.

Chapter 3 studies the averaging GMM estimator that combines a conservative GMM estimator based on valid moment conditions and an aggressive GMM estimator based on both valid and possibly misspecified moment conditions, where the weight is the sample analog of an infeasible optimal weight. We establish asymptotic theory on uniform approximation of the upper and lower bounds of the finite-sample risk difference between two estimators, which is used to show that the averaging estimator uniformly dominates the conservative estimator by reducing the risk under any degree of misspecification. Extending seminal results on the James-Stein estimator, the uniform dominance is established in non-Gaussian semiparametric nonlinear models. The simulation results support our theoretical findings.

Chapter 4 examines properties of permutation tests in the context of synthetic control. Permutation tests are frequently used method of inference for synthetic control when the number of potential control units is small. We show that the size of permutation tests may be distorted. Several alternative methods are discussed.