Sophistication of the Naive Bootstrap

朴素引导程序的复杂性

• 批准号: 0525707
• 负责人: Timothy Vogelsang
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0525707&HistoricalAwards=false
• 关键词: Sophistication; Naive; Bootstrap

In the bootstrap literature for dependent data there is widespread agreement on the properties of the block bootstrap when applied to tests based on HAC robust variance estimators. Several of these properties include: i) the i.i.d. bootstrap does not work when data are dependent, ii) the naive bootstrap, defined as a bootstrap where the formula used in the bootstrap world is the same as the formula used to compute the statistic using the actual data, is no more accurate than the usual first order asymptotic approximation and iii) the bootstrap for the Bartlett kernel based test is no more accurate than the standard first-order asymptotic approximation, whereas other kernels (including the quadratic spectral kernel) can lead to bootstrap tests more accurate than the standard asymptotic tests. In a recent paper, the investigator reported small sample simulation results for HAC robust tests in a simple location model that cast doubt on this conventional wisdom. It was found that i) the naive bootstrap, including the i.i.d. bootstrap, can dramatically outperform the standard normal approximation ii) this improvement occurs for many kernels including the Bartlett kernel and iii) the naive block bootstrap closely follows the PI's recently developed fixed-bandwidth (fixed-b) asymptotic approximation. The striking differences between patterns observed in small samples and those predicted by the standard theoretical results are puzzling. The purpose of this project is to develop a theoretical framework that can explain these bootstrap puzzles. The approach is to develop higher order asymptotic expansions within the fixed-b asymptotic framework. Progress is possible using recently developed expansions for partial sums of stationary time series due to Park (2003). Preliminary results suggest that the block bootstrap has the same first order fixed-b asymptotic term. If it can be shown that fixed-b asymptotics is theoretically more accurate than the standard asymptotic approximation, a new benchmark by which to assess the bootstrap will be established. Whether the block bootstrap is systematically more accurate than fixed-b asymptotics is an important topic of this proposal. Simulations suggest this may be true. Broader Impacts: The bootstrap has become a widely used and useful tool in statistics and econometrics. It is a flexible and convenient way of obtaining critical values for hypothesis tests. Although computationally intensive, the bootstrap has become much easier to implement due to the speed of modern computers. From a practical standpoint, the bootstrap is appealing because in many cases it can deliver more accurate approximations than conventional asymptotic theory. In other words, bootstrap critical values are often more accurate than asymptotic critical values. It is useful for empirical researchers to know when to expect the bootstrap to perform well in practice. The goal of the research in this project is to develop a new theoretical framework for assessing the performance of the bootstrap when applied to heteroskedasticity autocorrelation robust (HAC) test statistics in time series models. The impetus for developing this new framework is the tendency of existing theoretical approaches, e.g. the Edgeworth expansion approach, to understate the usefulness of the bootstrap in practice. The new approach is based on expansions for partial sums of stationary random variables and preliminary results suggest the new approach can explain the often superior performance of the bootstrap over standard asymptotics in HAC robust testing. While the new theory is being developed specifically for HAC robust tests, the approach could lead to new ways of understanding the bootstrap for other testing problems.

在相关数据的引导文献中，当应用于基于 HAC 鲁棒方差估计器的测试时，块引导的属性得到了广泛的一致。其中几个属性包括： i) i.i.d.当数据相关时，引导程序不起作用，ii) 朴素引导程序（定义为引导程序，其中引导程序世界中使用的公式与使用实际数据计算统计量的公式相同）不比通常的一阶渐近近似更准确，以及 iii) 基于 Bartlett 核的测试的引导程序不比标准一阶渐近更准确 近似，而其他内核（包括二次谱内核）可以导致引导测试比标准渐近测试更准确。在最近的一篇论文中，研究人员报告了在简单位置模型中进行 HAC 稳健测试的小样本模拟结果，这对这种传统观点提出了质疑。结果发现 i) 朴素的引导程序，包括 i.i.d. bootstrap，可以显着优于标准正态近似 ii) 这种改进发生在许多内核中，包括 Bartlett 内核，以及 iii) 朴素块引导程序紧密遵循 PI 最近开发的固定带宽 (fixed-b) 渐近近似。小样本中观察到的模式与标准理论结果预测的模式之间的显着差异令人费解。该项目的目的是开发一个可以解释这些引导难题的理论框架。该方法是在固定 b 渐近框架内开发高阶渐近展开式。由于 Park (2003)，使用最近开发的固定时间序列部分和的扩展是可能取得进展的。初步结果表明块自举具有相同的一阶固定 b 渐近项。如果可以证明固定 b 渐近在理论上比标准渐近近似更准确，则将建立一个评估自举的新基准。块引导是否在系统上比固定 b 渐近更准确是该提案的一个重要主题。模拟表明这可能是真的。 更广泛的影响：引导程序已成为统计和计量经济学中广泛使用且有用的工具。它是获取假设检验临界值的一种灵活且方便的方法。尽管计算量很大，但由于现代计算机的速度，引导程序变得更容易实现。从实用的角度来看，自举法很有吸引力，因为在许多情况下它可以提供比传统渐近理论更准确的近似值。换句话说，自举临界值通常比渐近临界值更准确。对于实证研究人员来说，了解何时期望引导程序在实践中表现良好非常有用。该项目的研究目标是开发一个新的理论框架，用于评估将引导程序应用于时间序列模型中的异方差自相关稳健（HAC）测试统计时的性能。发展这个新框架的动力是现有理论方法的趋势，例如Edgeworth 扩展方法，低估了引导程序在实践中的有用性。新方法基于平稳随机变量部分和的扩展，初步结果表明新方法可以解释在 HAC 鲁棒测试中引导程序通常优于标准渐近的性能。虽然新理论是专门针对 HAC 稳健测试而开发的，但该方法可能会带来理解其他测试问题的引导程序的新方法。