Sophistication of the Naive Bootstrap

在相关数据的自举法文献中，对于块自举法应用于基于HAC鲁棒方差估计的检验时的性质有广泛的共识。这些属性包括：i) i.i.d自举在数据依赖时不起作用，ii)朴素自举，定义为自举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的自举，并不比通常的一阶渐近近似更准确，iii)基于Bartlett核的自举测试并不比标准的一阶渐近近似更准确。而其他核（包括二次谱核）可以导致自举测试比标准渐近测试更准确。在最近的一篇论文中，研究者报告了在一个简单的位置模型中进行HAC鲁棒性测试的小样本模拟结果，对这种传统智慧提出了质疑。发现i)朴素引导，包括i.i.d引导，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内核和iii)朴素块引导密切遵循PI最近开发的固定带宽（固定b）渐近近似。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显著差异令人困惑。这个项目的目的是发展一个理论框架，可以解释这些引导谜题。该方法是在固定b渐近框架内发展高阶渐近展开式。由于Park(2003)，使用最近开发的平稳时间序列部分和的展开式可以取得进展。初步结果表明，块bootstrap具有相同的一阶固定b渐近项。如果可以证明固定b渐近理论比标准渐近近似更精确，则可以建立一个评估自举的新基准。块自举是否比固定b渐近更精确是本文的一个重要主题。模拟表明这可能是真的。更广泛的影响：自举已经成为统计学和计量经济学中广泛使用和有用的工具。这是一种灵活而方便的方法来获得假设检验的临界值。虽然计算密集，但由于现代计算机的速度，引导已经变得更容易实现。从实际的角度来看，自举法很有吸引力，因为在许多情况下，它可以比传统的渐近理论提供更准确的近似。换句话说，自举临界值通常比渐近临界值更准确。对于经验研究者来说，知道什么时候期望自举在实践中表现良好是有用的。本项目的研究目标是开发一个新的理论框架，用于评估自举在时间序列模型中应用于异方差自相关鲁棒性（HAC）检验统计量时的性能。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，低估了自举在实践中的有用性。新方法基于平稳随机变量部分和的展开式，初步结果表明，新方法可以解释在HAC鲁棒性检验中自举优于标准渐近的性能。虽然新理论是专门为HAC鲁棒性测试开发的，但这种方法可能会导致理解其他测试问题的自举的新方法。

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.