Optimal Bandwidth Choice for Hypothesis Testing and Interval Estimation in GMM Regression

自相关是时间序列数据中的一种常见现象。在存在自相关的情况下，回归参数的普通最小二乘（OLS）估计在一般情况下仍然是相合的。然而，通常的标准误差公式是无效的。在计量经济学中，许多实用的方法利用异方差和自相关一致标准误来获得稳健的推断。这些标准误差现在广泛应用于统计软件包中。实际的问题是，从业者需要选择所谓的带宽，在构建这些标准误差的调谐参数。调谐参数是很重要的，因为不同的调谐参数可能会导致定性不同的结论。现有的带宽选择方法都是基于最小化相关非参数量的均方误差（MSE）准则，在这种情况下，这是长期方差估计。这样的带宽选择被设计为在MSE意义上对于长期方差的点估计是最优的，但不一定最适合于假设检验和置信区间估计。本计画提出以最小化准则函数或损失函数之频宽选择，而准则函数或损失函数系针对假设检定与区间估计。对于假设检验，损失函数是I类错误（错误拒绝的概率）和II类错误（错误接受的概率）的加权平均值。对于区间估计，损失函数被认为是覆盖概率的误差，即真实覆盖概率与标称覆盖概率之间的差。因此，新提出的带宽选择规则解决了假设检验和区间估计中的核心问题。半参数检验和区间估计的最优带宽选择是时间序列回归中的一个长期问题。制定最佳选择程序并不简单，并且涉及一些概念和技术挑战。本项目面对这一挑战，提出了一个方法，在理论上健全和经验相关。 该项目大大推进了当前时间序列研究的前沿。 初步计算表明，为了优化新准则，人们将选择带宽来平衡长期方差估计量的渐近偏差和方差。这与传统的MSE标准形成鲜明对比，该标准平衡了平方渐近偏差与方差。一些有限的模拟表明，所提出的方法是有前途的。广泛影响： 该项目将对时间序列数据的理论和实践分析产生重大而深远的影响。虽然该理论框架是专门为异方差、自相关稳健检验和区间估计而建立的，但其思想和方法可用于一般非参数和半参数模型中调整参数的最优选择。调整参数可以是核平滑中的带宽或筛近似中的项数。从实际的角度来看，新的带宽选择规则有可能制定一个标准的实践计算的自相关鲁棒标准误差。社会科学和自然科学的研究人员可以使用新开发的带宽选择规则来执行更精确和可靠的推理。这将改善相关领域的实证研究。

Autocorrelation is a common phenomenon in time series data. In the presence of autocorrelation, the ordinary least squares (OLS) estimator of regression parameters is still consistent in general. However, the usual formulae for standard errors are invalid. Many practical methods in econometrics make use of heteroskedasticity and autocorrelation consistent standard errors in order to obtain robust inferences. These standard errors are now widely implemented in statistical packages. The practical problem is that the practitioner is required to choose the so-called bandwidth, a tuning parameter in constructing these standard errors. The tuning parameter is important because different tuning parameters may lead to qualitatively different conclusions.Existing methods for bandwidth selection are all based on minimizing the mean square error (MSE) criterion of the relevant nonparametric quantity, which in this context is the long-run variance estimator. Such a choice of the bandwidth is designed to be optimal in the MSE sense for point estimation of the long run variance, but is not necessarily best suited for hypothesis testing and confidence interval estimation. This project proposes to choose the bandwidth to minimize a criterion function or loss function that is directed at hypothesis testing and interval estimation. For hypothesis testing, the loss function is taken to be a weighted average of the type I error (the probability of false rejection) and the type II error (the probability of false acceptance). For interval estimation, the loss function is taken to be the error in coverage probabilities, i.e. the difference between the true coverage probability and the nominal coverage probability. The newly proposed bandwidth choice rules thus address the central concerns of interest in hypothesis testing and interval estimation.Optimal bandwidth selection for semiparametric testing and interval estimation is a long-standing problem in time series regressions. Developing an optimal selection procedure is not straightforward and involves some conceptual as well as technical challenges. The present project confronts this challenge by proposing an approach that is theoretically sounded and empirically relevant. This project significantly advances the frontiers of current time series research. Preliminary calculations show that, in order to optimize the new criteria, one would choose the bandwidth to balance the asymptotic bias and variance of the long run variance estimator. This is in sharp contrast with the conventional MSE criterion that balances the squared asymptotic bias with variance. Some limited simulations show that the proposed approach is promising. Broad Impacts: This project will have significant and far-reaching impacts on both theoretical and practical analyses of time series data. While the theoretical framework is developed specifically for heteroskedasticity and autocorrelation robust test and interval estimation, the idea and approach can be used to optimally select the tuning parameterin general nonparametric and semiparametric models. The tuning parameter can be the bandwidth in kernel smoothing or the number of terms in sieve approximation. From a practical perspective, the new bandwidth selection rule has the potential for developing a standard of practice for the computation of autocorrelation robust standard errors. Researchers in social sciences and natural sciences may use the newly developed bandwidth choice rule to perform more precise and reliable inferences. This will improve empirical studies in related fields.