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.
