EXISTENCE AND CHARACTERIZATION OF CONDITIONAL DENSITY PROJECTIONS
EXISTENCE AND CHARACTERIZATION OF CONDITIONAL DENSITY PROJECTIONS
复制标题
条件密度投影的存在和表征
作者：
Ivana Komunjer;Giuseppe Ragusa
本文提出了条件密度在由条件矩约束定义的集合上的投影存在且唯一的基本条件。并给出了得到的投影的解析表达式。使用条件密度投影的应用范围很广。导出的结果在许多领域都有潜在的用途，包括：半参数有效估计和（条件）矩模型的最佳测试，半参数模型的贝叶斯先验确定和推理，密度预测和基于模拟的计量经济学分析。关于存在性，我们提出了三种不同的假设组合，这些假设组合都足以证明投影存在并且是唯一的。所提出的条件在对条件密度之间的散度的限制和对定义投影集的矩函数的限制之间表现出明显的权衡。根据应用程序的性质，研究人员可以选择使用哪一组条件。我们的第二组结果描述了投影的特征。投影密度的表达式是新的，尽管考虑到之前获得的无条件情况的结果并不令人惊讶。该投影具有原投影问题的对偶特征。然而，在建立强对偶性时，我们使用的约束条件比Borwein和Lewis （1991a, 1992a, 1993）在他们关于无条件情况的开创性工作中使用的约束条件弱。
In this paper we propose primitive conditions under which a projection of a conditional density onto a set defined by conditional moment restrictions exists and is unique. Moreover, we provide an analytic expression of the obtained projection. The range of applications where conditional density projections are used is wide. The derived results are potentially useful in a variety of areas including: semiparametric efficient estimation and optimal testing in (conditional) moment models, Bayesian prior determination and inference in semiparametric models, density forecasting, and simulation-based econometric analysis. Regarding existence, we propose three different combinations of assumptions that are all sufficient to show that the projection exists and is unique. The proposed conditions exhibit a clear trade off between restrictions put on the divergence between the conditional densities and on the moment function which defines the projection set. Depending on the nature of the application, the researcher can pick and choose which set of conditions to use. Our second set of results characterizes the projection. The expression for the projected density is new though not surprising given the previously obtained results for the unconditional case. The projection is characterized by the dual of the original projection problem. In establishing the strong duality, however, we work with a constraint qualification condition that is weaker than that used by Borwein and Lewis (1991a, 1992a, 1993 in their seminal work concerning the unconditional case.