Essays on asymptotic methods in econometrics

Abstract

This thesis consists of three essays that contribute to statistical methods in econometrics. Chapter 1 develops new theory of integrable empirical processes and applies it to outlier robustness analysis. A frequent concern in empirical research is to ensure that a handful of outlying observations have not driven the key empirical findings. This chapter constructs a formal statistical test of outlier robustness and provides its theoretical foundation. The key is to observe that statistics related to outlier robustness analyses are represented as L-statistics--integrals of empirical quantile functions with respect to sample selection measures-and to consider these elements in appropriate normed spaces. We characterize the asymptotic distribution of L-statistics and prove the validity of nonparametric bootstrap. An empirical application shows the utility of the proposed test. Chapter 2 establishes the theory of weak identification in semiparametric models and provides an efficiency concept for weakly identified parameters. We first formulate the defining feature of weak identification as weak regularity, the asymptotic dependence of a parameter on the model score. While this feature deems consistent and equivariant estimation of a weakly regular parameter impossible, we show that there exists an underlying parameter that is regular and fully characterizes the weakly regular parameter. Using the minimal sufficient underlying regular parameter, we define weak efficiency for a weakly regular parameter through local asymptotic Rao-Blackwellization. Simulation shows that efficiency of popular estimators in linear IV models can be improved under heteroskedasticity. Chapter 3 provides a method to account for estimation error in financial risk control. As accuracy of estimated risk is subject to estimation error, risk control based on estimated risk may fail to control the true, unobservable risk. We show that risk measures that give bounds to the probabilities of bad events can be effectively controlled by the Bonferroni inequality when the distributions of their estimators are known or estimable. We call such risk measures tail risk measures and show that they subsume Value-at-Risk and expected shortfall. An empirical application to portfolio risk management shows that a multiplier of 1.3 to 1.9 can control the true risk probability of expected shortfall at 10%.