Evaluation and Comparison of Econometric Models Using Nonparametric Likelihood and Bootstrap

本项目涵盖计量经济模型规格评估和比较中的以下三个主题：在预测推理中使用自举平滑，在比较可能性错定的动态计量经济模型中使用非参数方法，以及根据Hoeffding的渐近相对效率度量的经验似然比检验的最佳特性。这些主题是计量经济模型评估和比较的主要研究活动的核心。本项目对计量经济学理论和应用计量经济学有重大贡献。第一个项目讨论预测推理，它可以看作是一个使用某种数据分割方案的验证问题。为了对经济计量模型进行预测测试，指定损失函数，并将数据集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，根据验证集测试估计的模型。因此，预测试验的有效性取决于风险估计器的准确性。传统的风险估计是在估计参数值处评估的损失函数值的验证样本平均值。这个过程可以看作是前向交叉验证的一种变体，因此称为前向验证（FV）。FV的一个缺点是它没有明确地将参数估计的不确定性纳入风险估计。本项目建议使用自举平滑（BS）来弥补这一缺点。BS算法重复以下两步：(1)通过对训练集(2)重新采样获得未知参数的bootstrap draw，在FV中，计算损失函数值的验证-样本平均值，但这次在(1)的bootstrap draw处进行评估。在将这两个步骤应用于（足够多的）bootstrap绘图之后，取验证样本平均值的平均值。这种方法类似于Efron的“留一个bootstrap”或Breiman的“bagging”，它们对于不连续的损失函数是有效的。第一个项目发展了以下定理：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，从而提供了一个清晰且潜在可观的渐近效率增益。这是一个相当令人惊讶的结果，但与“留一个”方法和装袋的良好实际性能相一致。讨论了与贝叶斯方法的一些联系。BS方法应用于Merton的市场时机可预测性度量和其他经验例子。第二个项目涉及可能指定错误和可能未嵌套的力矩条件模型的比较。力矩条件模型的标准规范试验假设在零点下存在完全正确的力矩限制，但有时这种假设在实践中是不合理的。即便如此，根据拟合优度的总体衡量标准来比较这些可能被错误指定的模型仍然是有趣的。本项目通过结合非参数似然法和Vuong（参数）模型比较检验，开发了一种用于这种比较的装置。此外，利用非参数似然的局部光滑版本将该方法扩展到比较可能存在错误指定的条件矩约束模型。第三个项目考虑力矩条件模型。它追求对过度识别限制的最佳测试，这通常由汉森的“j测试”进行测试。最近，使用非参数似然的一些版本开发了汉森检验的几种替代方法。虽然传统的局部渐近效率比较无法区分这些相互竞争的测试，但Hoeffding最初提出的全局效率度量使得有可能建立IID样本的经验似然比测试的最优性。还研究了将这种分析扩展到相关观测的方法。

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.