Improved Semi-Nonparametric Estimation and Testing By Modified Likelihood

经济理论通常对进入经济模型的组件的具体功能形式很少提及。这导致经济学中越来越多地使用非参数和半参数估计和测试方法，因为这些方法通常对感兴趣的模型施加较弱的功能限制。许多流行的非参数和半参数估计都涉及到所谓的核平滑。然而，这些实现起来可能很有挑战性，因为它们涉及到选择合适的带宽，这是估计器的一个组成部分：最终的估计器通常对带宽选择很敏感。不幸的是，理论对如何在有限样本中选择这些提供了很少的指导：首先，带宽不会出现在参数估计量的渐近分布中。其次，为了使第一步估计误差以最优速率消失，需要进行欠平滑。这就排除了标准的带宽选择方法，比如插件和交叉验证。对于一些特殊的估计器，已经开发了一些方法，但是这些方法实现起来很复杂，并且并不总是执行得很好。我们在这里提出了一类新的半参数轮廓估计器，它不会遭受这些问题的困扰：我们开发了定义估计器的标准目标（似然）函数的修改版本。这种改进使得它既可以用来估计非参数分量，也可以用来估计参数分量。这种修改的优点有三个方面：首先，修改确保了通常出现在参数估计器展开中的误差项现在消失了。因此，我们期望修改后的版本将具有更好的有限样本性能。其次，通过去除误差项，我们不必为了使第一步估计误差以最优速率消失而进行欠平滑。因此，可以使用标准带宽选择方法。最后，所提出的改进估计器并不比标准估计器更难实现，并且不需要大量的计算。我们还演示了如何使用改进的目标函数来改进使用核平滑方法的现有非参数和半参数测试程序。根据Pitman的相对效率标准，改进后的测试优于原始测试，因此更强大。与基于核的半参数估计过程一样，如何在实现基于核的测试过程中选择带宽的问题在很大程度上尚未解决。我们将研究在这个项目中开发的新一类测试统计量的带宽选择问题。新方法可用于改进现有的半参数模型的计量经济学方法，包括部分线性模型、单指标模型、（半）变系数模型和具有时变参数的模型。这些模型和许多其他模型将在项目中考虑。

Economic theory often has little to say about the specific functional forms of components entering economic models. This has lead to an increased use of non- and semiparametric estimation and testing methods in economics since these in general impose weaker functional restrictions on the models of interest.Many popular non- and semiparametric estimators involve so-called kernel-smoothing. However, these can be challenging to implement since they involve choosing appropriate bandwidths which are an integral part of the estimators: The resulting estimators are in general sensitive to the bandwidth choice. Unfortunately, theory offers few guidelines for how these should be chosen in finite samples: First of all, the bandwidth does not appear in the asymptotic distribution of the parametric estimator. Secondly, for the first-step estimation error to vanish at an optimal rate, undersmoothing is required. This rules out standard bandwidth selection methods such as plug-in and cross-validation. For a few special estimators, methods have been developed, but these can be complicated to implement and do not always perform well.We here propose a novel class of semiparametric profile estimators that do not suffer from these problems: We develop a modified version of the standard objective (likelihood) function defining the estimator. The modification entails that it can be used to estimate both the nonparametric component and the parametric one. The advantages of this modification are three-fold: First, the modification ensures that an error term normally appearing in the expansion of the parametric estimator now vanishes. Thus we expect that the modified version will have better finite-sample properties. Second, by removing the error term, we do not have to undersmooth in order for the first-step estimation error to vanish at an optimal rate. Hence standard bandwidth selection methods can be used. Finally, the proposed modified estimator is no more difficult to implement than standard estimators and require no heavy computations.We also demonstrate how the modified objective function can be used to improve on existing non- and semiparametric testing procedures using kernel-smoothing methods. The modified tests are shown to dominate the original ones in terms of Pitman's relative efficiency criterion and as such are more powerful. As with the kernel-based semiparametric estimation procedures, the issue of how to select bandwidths in the implementation of kernel-based testing procedures is to a large extent unresolved. We will examine the issue of bandwidth selection for the new class of test statistics developed in this project.The novel procedures can be used to improve upon existing econometric methods developed for many semiparametric models, including partially linear models, single-index models, (semi-)varying-coefficient models, and models with time-varying parameters. These and many other models will be considered in the project.