Evaluation and Comparison of Econometric Models Using Nonparametric Likelihood and Bootstrap

本项目包括以下三个主题的评价和比较的计量经济模型规格：使用自举平滑的预测推理，使用非参数方法比较可能性错误指定的动态计量经济模型，和经验似然比检验的最佳性能的Hoeffding的渐近相对效率的措施。 这些主题是在计量经济模型评价和比较的主要研究活动的中心。 该项目将对计量经济学理论和应用计量经济学做出重大贡献。第一个项目讨论预测推理，它可以被视为使用某种数据分裂方案的验证问题。为了进行经济计量模型的预测测试，指定损失函数，并将数据集划分为训练集和验证集。使用训练集估计模型的未知参数，并使用预期损失（有时称为风险）作为标准，针对验证集测试估计的模型。因此，预测检验的有效性取决于风险估计器的准确性。 传统的风险估计量是在估计的参数值处评估的损失函数值的验证样本平均值。这个过程可以被看作是交叉验证的一个变体，它是向前执行的，因此称为向前验证（FV）。FV的缺点是没有将参数估计的不确定性明确地纳入风险估计中，本计画提出使用自助平滑法（BS）来弥补这个缺点。BS算法重复以下两个步骤：（1）通过像FV中那样重新搜索训练集（2）来获得未知参数的自举提取，计算损失函数值的验证样本平均值，但这次在自举提取时从（1）进行评估。在将这两个步骤应用于（足够多的）自举抽取后，取验证样本平均值的平均值。这种方法类似于Efron的“leave-one-out-bootstrap”或Breiman的“bagging”，已知它们对不连续损失函数有效。第一个项目开发了以下定理：自举平滑消除了大样本中指标损失函数的参数不确定性的影响，从而提供了一个明确的和潜在的渐近效率增益。这是一个相当令人惊讶的结果，但与“留一法”和装袋法的良好实际性能一致。与贝叶斯方法的一些连接进行了讨论。BS方法被应用到默顿的市场时机的可预测性措施和其他实证examinations.The第二个项目是关注可能被误指定和可能非嵌套的时刻条件模型的比较。矩条件模型的标准规范检验假设在零值下存在完全正确的矩约束，但有时这种假设在实践中是不合理的。即便如此，它仍然是感兴趣的比较这种可能被错误指定的模型在整体措施的拟合优度。本项目通过结合非参数似然方法和Vuong（参数）模型比较检验开发了一种用于这种比较的设备。此外，我们还利用局部平滑的非参数似然函数，将该方法推广到比较条件矩约束模型中可能出现的错误。第三个项目考虑矩条件模型。它追求过度识别限制的最佳测试，通常通过汉森的“J-检验”进行测试。“最近几个替代汉森的测试已经开发出使用一些版本的非参数似然。虽然传统的局部渐近效率比较不能区分这些竞争的测试，Hoeffding最初提出的全球效率措施，使人们有可能建立一个最优性的经验似然比测试IID样本。扩展此分析相关的意见也进行了研究。

This project covers the following three topics in evaluation and comparison of econometric model specifications: use of bootstrap smoothing in predictive inference, use of nonparametric methodology in comparing possibility misspecified dynamic econometric models, and optimal properties of empirical likelihood ratio tests in terms of Hoeffding's measure of asymptotic relative efficiency. These topics are central to major research activity in econometric model evaluation and comparison. This project should make major contributions to econometric theory and applied econometrics.The first project discusses predictive inference, which can be regarded as a validation problem that uses a certain data splitting scheme. To carry out a predictive test of an econo-metric model, a loss function is specified and the data set is divided into the training set and the validation set. Unknown parameters of the model are estimated using the training set, and the estimated model is tested against the validation set, using the expected loss (sometimes called risk) as a criterion. Thus the efficacy of the predictive test depends on the accuracy of the risk estimator. A conventional risk estimator is the validation-sample average of loss function values eval-uated at the estimated parameter value. This procedure can be viewed as a variant of cross validation that is carried out forward, therefore called forward validation (FV). A drawback of FV is that it does not explicitly incorporate parameter estimation uncertainty into risk estimation.This project proposes to use bootstrap smoothing (BS) to remedy this drawback. The BS algorithm repeats the following two steps: (1) a bootstrap draw of an unknown parameter is obtained by resampling the training set (2) as in FV, the validation-sample average of the loss function values is calculated, but this time evaluated at the bootstrap draw from (1). After ap-plying these two steps to (sufficiently many) bootstrap draws, the average of validation-sample averages is taken. This method is similar to Efron's "leave-one-out-bootstrap," or Breiman's "bagging," which are known to be effective for discontinuous loss functions. The first project develops the following theorem: bootstrap smoothing eliminates the effect of parameter uncer-tainty for indicator loss functions in large samples, thereby providing a clear and potentially substantial asymptotic efficiency gain. This is a rather surprising result, but consistent with the good practical performance of "leave-one-out" methods and bagging. Some connections with Bayesian methods are discussed. The BS methodology is applied to Merton's predictability measure of market timing and other empirical examples.The second project is concerned with the comparison of possibly misspecified and possibly non-nested moment condition models. Standard specification tests for moment condition models assume the presence of an exactly correct moment restriction under the null, but sometimes this assumption is not reasonable in practice. Even so, it is still of interest to compare such possibly misspecified models in terms of an overall measure of goodness-of-fit. This project develops a device for this comparison by combining a nonparametric likelihood method and Vuong's (parametric) model comparison test. Furthermore, a local smoothed version of nonparametric likelihood is used to extend the method to comparing possibly misspecified conditional moment restriction models.The third project considers moment condition models. It pursues optimal tests of overidentifying restrictions, which are typically tested by Hansen's "J-test." Recently several alternatives to Hansen's test have been developed using some versions of nonparametric likelihood. While conventional local asymptotic efficiency comparisons cannot distinguish among these competing tests, a global efficiency measure originally proposed by Hoeffding makes it possible to establish an optimality property of empirical likelihood ratio tests for IID samples. Extensions of this analysis to dependent observations are also investigated.