Improved Semi-Nonparametric Estimation and Testing By Modified Likelihood

对于进入经济模型的组成部分的具体功能形式，经济学理论往往没有什么可说的。这导致了经济学中越来越多地使用非参数和半参数估计和检验方法，因为这些方法通常对利益模型施加较弱的函数限制。许多流行的非参数和半参数估计涉及所谓的核平滑。然而，这些方法的实施可能具有挑战性，因为它们涉及选择适当的带宽，而这些带宽是估计器的组成部分：所得到的估计器通常对带宽选择敏感。不幸的是，理论几乎没有提供在有限样本中如何选择这些参数的指导方针：首先，带宽没有出现在参数估计器的渐近分布中。其次，为了使第一步估计误差以最优速度消失，需要进行欠平滑。这排除了标准带宽选择方法，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，但这些方法实现起来可能很复杂，而且并不总是执行得很好。在这里，我们提出了一类新的半参数轮廓估计器，它们不存在这些问题：我们提出了定义估计器的标准目标(似然)函数的修改版本。这种修正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：首先，修改确保了通常出现在参数估计器的展开中的误差项现在消失了。因此，我们预计修改后的版本将具有更好的有限样本性质。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佳速率消失而进行欠平滑。因此，可以使用标准的带宽选择方法。最后，提出的修正估计器不比标准估计器更容易实现，并且不需要大量的计算，我们还展示了如何使用修正的目标函数来改进现有的使用核平滑方法的非参数和半参数检验方法。结果表明，改进后的检验在皮特曼相对效率准则方面优于原检验，因而更有说服力。与基于核的半参数估计程序一样，在基于核的测试程序的实现中如何选择带宽的问题在很大程度上是没有解决的。我们将研究这个项目中开发的新一类检验统计量的带宽选择问题。新的方法可以用于改进现有的针对许多半参数模型的计量经济学方法，包括部分线性模型、单指数模型、(半)变系数模型和具有时变参数的模型。这些和许多其他型号将在该项目中考虑。

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.