Sophistication of the Naive Bootstrap

朴素引导程序的复杂性

• 批准号: 0731413
• 负责人: Timothy Vogelsang
• 金额: $ 14.73万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-30 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0731413&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.

在有关依赖数据的引导文献中，当基于HAC稳健方差估计器应用于测试时，就块引导程序的属性有广泛的一致性。其中几个属性包括：i）I.I.D. bootstrap在数据取决于数据时不起作用，ii）幼稚的引导程序，定义为引导程序，其中引导世界中使用的公式与用于使用实际数据计算统计量的公式相同，并不比通常的第一阶近似近似值和iii更准确，而不是标准的bartlett tectertip the Bootstrate bartlett abartlett abartlett kernett kernelt kernelt kernelt kernelt kernelt kernelt kernelt kernel kernel kernel kernel tectertipt近似值，而其他内核（包括二次光谱核）可以比标准渐近测试更准确地导致引导测试。在最近的一篇论文中，研究者报道了在一个简单的位置模型中对HAC稳健测试的样本仿真结果，该模型对这种传统智慧产生了怀疑。发现我）幼稚的引导程序，包括I.I.D.引导程序，可以极大地胜过标准正常近似值ii）ii）对于包括Bartlett内核和III在内的许多内核进行了这种改进。在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明显差异令人困惑。该项目的目的是开发一个可以解释这些引导程序难题的理论框架。该方法是在固定B渐近框架内发展高阶渐近扩展。对于由于Park（2003）而导致的部分固定时间序列的部分资金，可以使用最近开发的扩展。初步结果表明，Block Bootstrap具有相同的一阶固定B渐近项。如果可以证明，固定B渐近剂在理论上比标准渐近近似更准确，则将建立一种评估引导程序的新基准。 Block Bootstrap是否比固定B渐近学更为准确，这是该提案的重要主题。模拟表明这可能是正确的。 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。这是获得假设检验关键值的灵活和方便的方法。尽管在计算密集程度上，但由于现代计算机的速度，引导程序变得更加容易实施。从实际的角度来看，引导程序很有吸引力，因为在许多情况下，它可以比常规渐近理论提供更准确的近似值。换句话说，bootstrap临界值通常比渐近临界值更准确。对于经验研究人员来说，知道何时可以期望引导程序在实践中表现良好。该项目中的研究的目的是开发一个新的理论框架，以评估在时间序列模型中应用于HeteroSkedasticity自相关鲁棒（HAC）测试统计量时，以评估引导程序的性能。开发这种新框架的动力是现有理论方法的趋势，例如Edgeworth的扩展方法低估了实践中引导程序的实用性。新方法是基于固定随机变量的部分总和的扩展，初步结果表明，新方法可以解释自举在HAC鲁棒测试中通常比标准渐近学的表现优越。尽管新理论是专门针对HAC鲁棒测试开发的，但该方法可能会导致新的方法来理解其他测试问题的引导程序。