Three Projects in Econometric Theory

经济学和社会科学中的大多数实证方法的有效性都来自于对它们在非常大的样本中如何表现的思想实验。隐含的希望是，确保在大样本中的良好性能，也会导致在适度小的数据集中获得可接受的性能。然而，对于一些推理问题，众所周知，标准的大样本近似太不准确，不能作为经验工作中典型遇到的样本大小的可靠指南。作为一种建设性的补救措施，有时可以将原始问题嵌入到另一个更好地捕获小样本特征的大样本近似中。本项目旨在在三个经验相关的计量经济学问题中开发这种替代大样本近似：如何使用持续时间序列进行推理；如何解释线性回归中大量控制变量的存在；以及如何对极端事件的概率和性质进行推理。计量经济学理论的最新进展经常考虑导致不同形式的大样本近似的参数序列或调整参数值。突出的例子包括弱工具渐近性，局部到单位时间序列渐近性，其中最大的自回归根取的值更接近于1，以及带宽等于样本量的固定分数的异方差和自相关鲁棒推断。本研究在三个不同的推理问题中发展了类似的可选渐近性：第一个项目通过让p个自回归根以及p-1个MA根以适当的速率收敛到统一，推广了局部到统一模型。第二个项目研究在存在大量潜在控制的情况下关于线性回归系数的推断，其中控制系数已知满足特定的l2界，该l2界可以解释为控制结果回归中R^2的界。第三个项目涉及基于iid样本的尾部属性推断问题，在唯一的假设下，对于给定和固定的k，极值理论适用于最大k个观测值。

Most empirical methods in economics and the social sciences derive their validity from the thought experiment on how they would perform in very large samples. The implicit hope is that ensuring good performance in large samples leads to acceptable performance in moderately small data sets as well. For some inference problems, however, it is known that the standard large sample approximation is too inaccurate to be a reliable guide for the sample sized typically encountered in empirical work. As a constructive remedy, it is sometimes possible to embed the original problem in an alternative large sample approximation that better captures the small sample features. This project aims at developing such alternative large sample approximations in three empirically relevant econometric problems: How to conduct inference with persistent time series; how to account for the presence of a large number of control variates in a linear regression; and how to conduct inference about the probability and properties of extreme events. Recent advances in econometric theory often consider sequences of parameter or tuning parameter values that lead to a different form of large sample approximations. Prominent examples include weak instrument asymptotics, local-to-unity time series asymptotics where the largest autoregressive root takes on values ever closer to one, and heteroskedasticity and autocorrelations robust inference with a bandwidth equal to a fixed fraction of the sample size. This research develops similar alternative asymptotics in three distinct inference problems: The first project generalizes the local-to-unity model by letting p autoregressive roots, as well as p-1 MA roots, converge to unity at the appropriate rate. The second project studies inference about a linear regression coefficient in the presence of a large number of potential controls, where the control coefficients are known to satisfy a particular L_2 bound that can be interpreted as a bound on the R^2 in a regression of the outcome on the controls. The third project concerns the problem of inference about tail properties based on an i.i.d. sample, under the sole assumption that extreme value theory to hold for the largest k observations, for a given and fixed k.