Statistical inference in moderately high-dimensional endogenous regressions

Abstract

This dissertation addresses three classical inference problems related to the issue of endogeneity, utilizing modern statistical techniques, particularly random matrix theory. The statistical models of interests are typically situated in moderately high-dimensional settings, where the number of regressors increases proportionally to the sample size. The primary objective is to offer fresh insights into fundamental issues in econometrics using contemporary statistical methods, while establishing a connection between high-dimensional econometrics and random matrix theory. This approach has the potential to generate novel perspectives and methodologies for addressing more complex problems in the field. Furthermore, re-examining classical problems not only enhances our understanding of the historical progression of statistics but also allows us to rediscover valuable insights that may have been overlooked over time.

Chapter 2 discusses my work supervised by Professor Jianfeng Yao and Professor Chen Wang on inference problems in the context of many instrumental variables, where the number of instruments grows proportionally with the sample size. We explore the asymptotic distributions of the first-stage $F$ test statistic for weak instruments in the many instruments framework. In contrast to standard results, we establish the asymptotic normality of the test statistic under appropriate regularity conditions. Our proof is based on the joint CLT for sesquilinear forms from the random matrix literature. Building upon the established theory, we demonstrate that the classical first-stage $F$ test may suffer from distorted sizes. To address this issue, we propose a valid version by correcting its distribution.

Chapter 3 focuses on my joint work with Professor Jianfeng Yao and Professor Chen Wang on testing exogeneity in a doubly moderate dimensions setting, where both the number of regressors and instruments grow at a rate no larger than the sample size. We find that the most widely used Durbin-Wu-Hausman (DWH) test can be oversized. To overcome the limitation of the traditional DWH test, we propose a dimension-adaptive test statistic for no endogeneity based on the DWH test statistic. We establish its asymptotic properties and show that it possesses the correct size in both fixed and moderately high dimensions. We further extend the proposed test to settings with many exogenous regressors.

Chapter 4 delves into my joint work with Professor Jianfeng Yao and Professor Zhaoyuan Li on testing error cross-sectional independence in large panel data models. We propose a unified test procedure and its power enhancement version that show robustness for a wide class of panel model contexts. We establish the asymptotic validity of the test procedures under the simultaneous limit scheme where the number of time periods and the number of cross-sectional units go to infinity proportionally. The derived theories are accompanied by detailed Monte Carlo experiments, which confirm the robustness of the two tests and also suggest the validity of the power enhancement technique.