Unification of Bayesian and Frequentist Inference in Econometrics

在经验经济学中，有两种主要的方法来描述模型参数和预测的不确定性。经典方法，也被称为频率法，根据推理和估计过程在许多可能的数据集上的平均表现来评估它们。相比之下，贝叶斯方法关注给定的数据集，并对该数据集执行有条件的推理。贝叶斯(条件)和经典(频率)性质的可取性在经验经济学和更广泛的统计学文献中都是很好的理解的。在一大类标准估计问题中，经典和贝叶斯方法提供了大致相同的结果。因此，通常的经典估计方法具有吸引人的频率和条件性质。然而，近年来，经验经济学中的许多注意力都集中在非标准估计问题上，在这些问题上，两种方法之间的等价性可能会失败。例如，在高度持续性时间序列的模型中出现了非标准问题。许多经济时间序列，如通货膨胀和利率，具有很强的持久性。另一类重要的非标准问题包括部分或弱识别参数的问题，换句话说，数据只包含关于兴趣量的相对少量的信息。关于模型参数和预测的不确定性通常由集合估计器来描述，对于给定的数据，集合估计器提供关于兴趣量的一组可能的值。现有的构造集合估计量的经典方法不一定对非标准问题中的不确定性提供令人信服的描述，因为它们可能具有较差的条件性质。拟议研究的第一部分旨在开发一种方法，用于评估和构建非标准计量经济学问题中的集合估计器。在这个框架下，吸引集估计既具有频率性质又具有条件性质。所提出的方法包括理论结果和数值算法。它将通过经济学应用中经常出现的一些非标准问题来说明。另一个尚未完全理解经典方法和贝叶斯方法之间关系的重要领域是具有高维参数的灵活模型。这类模型在经济应用中很有用，因为它们通过施加较少的先验限制来“让数据说话”。这项研究的第二部分试图为如何构建既具有贝叶斯属性又具有经典属性的灵活模型的文献做出贡献。

There are two main approaches to describing uncertainty about model parameters and forecasts in empirical economics.The classical approach, which is also known as the frequentist approach, evaluates inference and estimation procedures in terms of how they perform on average over many possible datasets. In contrast, the Bayesian approach focuses on a given dataset and performs inference conditionally on this dataset. The desirability of both Bayesian (conditional) and classical (frequentist) properties is well understood in empirical economics and, more generally, in the statistics literature.In a large class of standard estimation problems, the classical and Bayesian approaches deliver approximately equivalent results. Thus, usual classical estimation procedures have attractive frequentist and conditional properties. In recent years, however, though, a lot of attention in empirical economics has been devoted to non-standard estimation problems, where the equivalence between the two approaches can fail. For instance, non-standard problems arise in models for highly persistent time series. Many economic time series such as inflation and interest rates are highly persistent. Another important class of non-standard problems includes problems with partially or weakly identified parameters, in other words, problems in which data contain only a relatively small amount of information about a quantity of interest.Uncertainty about model parameters and forecasts is usually described by set estimators, which for given data provide a set of likely values for a quantity of interest. Existing classical methods for construction of set estimators do not necessarily provide compelling descriptions of uncertainty in non-standard problems as they might have poor conditional properties. The first part of the proposed research intends to develop a methodology for evaluation and construction of set estimators in non-standard econometric problems. In this framework, attractive set estimators possess both frequentist and conditional properties. The proposed methodology includes theoretical results and numerical algorithms. It will be illustrated on a number of non-standard problems that routinely arise in economics applications.Another important area where the relationship between the classical and Bayesian approaches has not been completely understood is flexible models with high-dimensional parameters. Such models are useful in economic applications as they "let data speak" by imposing fewer a priori restrictions. The second part of the proposed research seeks to contribute to the literature on how to construct flexible models that posses both Bayesian and classical properties.