Essays in Nonparametric Econometrics

Abstract

This dissertation consists of three chapters on inference in various non- and semiparametric problems. The chapters are self-contained and can be read separately. Each chapter ends with an appendix that collects the proofs and technical details. <br/>

In the first chapter, we study inference on parameters of the form <em>ϕ</em>(<em>θ</em>₀), where <em>ϕ</em> is a known directionally differentiable transformation and <em>θ</em>₀ is an unknown parameter. We focus on settings, where <em>θ</em>₀ is an unknown function estimated using some nonparametric estimator <em>θ</em>_n. As many nonparametric estimators do not converge in distribution, existing extensions to the Delta method are not applicable in our setting. We propose to use strong approximations to the distribution of <em>θ</em>_n as an alternative concept to convergence in distribution. Further, we present a notion of directional differentiability which is sufficiently flexible to handle the irregularity of nonparametric estimators. These concepts enable us to derive a new Delta method which approximates the distribution of the plug-in estimator <em>ϕ</em>(<em>θ</em>_n). Since these distributional approximations are rarely pivotal, we suggest a simulation-based estimator and provide conditions for its consistency. Confidence intervals based on this estimator are shown to provide local size control under conditions on the directional derivative of <em>ϕ</em>. We illustrate the applicability of our results in two examples and study its finite sample performance in a simulation study. <br/>

Anti-concentration bounds play an important role in the modern theory on confidence intervals and testing in settings such as high-dimensional and nonparametric statistics. In the second chapter, we establish such a bound for sublinear and continuous functionals of tight Gaussian random vectors in real-valued Banach spaces. The bound is dimension-free and therefore equally applies to finite- as well as infinite-dimensional settings. It imposes only weak restrictions on the covariance structure of the Gaussian vectors. As an application of our anti-concentration bound, we derive Berry-Esseen type bounds for sublinear and continuous functionals of high-dimensional mean vectors and kernel-type estimators. <br/>

The last chapter, which is joint work with Michael Vogt, studies estimation and inference in the high-dimensional partially linear model <em>Y</em> = δ + <em>m</em>(<em>T</em>) + <em>X</em>^T β + ϵ, where <em>m</em> is a smooth unknown function and β a sparse unknown regression parameter. The dimension of the covariates <em>X</em> is allowed to increase with the sample size and in particular is allowed to be larger than the sample size. We propose an estimator of β which attains the same rates as the infeasible Lasso estimator which knows the unknown function <em>m</em>. Further, we show that ad-hoc estimators of <em>m</em> might be biased due to the estimation of the high-dimensional parameter β and propose an orthogonalized Nadaraya-Watson estimator of <em>m</em> which effectively decreases this high-dimensional bias. This estimator is shown to converge at the same rates as an infeasible Nadaraya-Watson estimator which knows the true value of β. Based on this estimator, we propose a test for the hypothesis that <em>m</em> = 0 which generalizes the idea of significance testing in linear models to allow for general nonlinear effects of <em>T</em> on <em>Y</em>. Moreover, we propose a consistent multiplier bootstrap in order to set the critical values and show uniform consistency of the resulting set against local Hölder balls. We study the finite sample performance of our proposed test in a simulation study and demonstrate its good debiasing and power properties.