Collaborative Research: Non-Standard Asymptotic Theory for Semiparametric Estimators

社会科学和自然科学中的现代统计和计量经济学模型非常复杂，通常包含许多未知参数。其中一些参数是研究人员和决策者特别感兴趣的（例如，治疗的平均效果），而其他则不是（例如，回归函数的精确形式或观测协变量的概率定律）。后者的参数通常被称为滋扰参数，因为它们的值是需要的，以便进行有效的统计推断的参数的利益，即使研究人员对他们不感兴趣。这些模型中的一类重要模型是所谓的半参数模型，其显着特征是讨厌的参数是函数（而不是数字）。在统计学和计量经济学中进行半参数模型推断的一种现代的、行之有效的方法是首先（使用可用数据）以灵活的、非参数的方式估计讨厌的参数，然后使用这些估计作为其真实值的初步猜测，以便对感兴趣的参数进行推断。这种方法通常被称为半参数推断，由于其灵活性和对模型错误设定产生的偏差不敏感，因此特别有用。半参数推断方法在理论研究人员中非常流行，部分原因是其良好且易于理解的大样本性质（假设大量数据的近似）。然而，这些推理程序在实证研究人员和政策制定者中相当不受欢迎，主要是因为他们被认为是高度敏感的方式，他们在实践中实施。具体而言，大多数半参数推断程序的一个重要缺点是，它们依赖于非参数技术的滋扰参数的估计，这又需要选择的调谐和平滑参数。这些额外的参数被人为地引入到推理过程中，以灵活地近似未知函数（讨厌的参数）。在文献中采用的大样本近似忽略了这些额外的参数的影响，这些额外的参数是人为引入的推理过程的建设。这一事实，反过来，导致一个重要的缺乏稳健性的半参数推断程序，也就是说，在选择的调整和平滑参数的小变化导致显着不同的经验结果，使应用工作不可靠的一般。换句话说，这种鲁棒性的缺乏通常转化为不正确的统计推断，这可能导致研究人员和政策制定者从采用这些半参数推断程序的实证工作中得出有缺陷的结论。替代大样本近似常用的半参数推断程序，（至少部分地）考虑了在推断过程中涉及的调整和平滑参数的特定用户定义的选择的效果。这种替代渐近理论导致更“鲁棒”的统计推断程序，因为它捕捉到了传统大样本近似所假设的某些项的影响。该项目将分两个主要阶段进行。首先，将针对特定的半参数例子，包括加权平均导数和部分线性模型，开发替代的大样本近似。不仅这些模型本身是有意义的，而且它们还将提供一些关键成分，以理解本提案中研究的非标准大样本近似所产生的新理论特征。除其他问题外，我们的目标是建立一个替代的一阶大样本分布，获得有效的标准误差估计，开发新的方法来选择的值的调整和平滑参数，研究常用的reservation程序的有效性，并探讨高阶的替代渐近近似的影响。一旦这些特殊的半参数过程的研究是很好地理解，调查的第二阶段将是发展概括和统一的理论结果概述的特殊的例子，这将涵盖许多其他问题的兴趣。这项研究的结果预计将有利于几个领域的研究，从经济学或政治学，生物统计学或公共卫生，使研究人员能够在半参数模型中进行“稳健”的推断，并使半参数推断对研究人员和决策者更具吸引力。为了进一步增加这项研究提案的影响，一个关键目标是为常用的平台提供计算机代码，并编写一份非技术调查报告，讨论理论和实施经典结果和从研究中出现的新结果。

Modern statistical and econometric models in social and natural sciences are complex and typically include many unknown parameters. Some of these parameters are of particular interest to the researchers and policymakers (e.g., the mean effect of a treatment), while others are not (e.g., the exact form of a regression function or the probability law of the observed covariates). The latter parameters are usually called nuisance parameters because their values are needed in order to conduct valid statistical inference on the parameters of interest, even though the researchers are not interested in them. An important class of these models are the so-called semiparametric models, which have the distinctive feature that the nuisance parameters are functions (as opposed to numbers). A modern, well-established approach in statistics and econometrics to conduct inference in semiparametric models is to estimate in a flexible, non-parametric way the nuisance parameters first (using the available data), and then employ these estimates as a preliminary guess of their true value in order to conduct inference on the parameters of interest. This procedure is generically referred to as semiparametric inference, and is particularly useful because of its flexibility and lack of sensitivity to biases generated by model misspecification.Semiparametric inference procedures are very popular among theoretical researchers, partially because of their nice and well understood large sample properties (approximations that assume a large amount of data). However, these inference procedures are considerably less popular among empirical researchers and policymakers, mainly because they are known to be highly sensitive to the way that they are implemented in practice. Specifically, an important drawback of most semiparametric inference procedures is that they rely on non-parametric techniques for the estimation of the nuisance parameters, which in turn require the selection of tuning and smoothing parameters. These additional parameters are artificially introduced in the inference procedure to flexibly approximate the unknown functions (the nuisance parameters). The large sample approximations employed in the literature ignore the effect of these additional parameters that are artificially introduced in the construction of the inference procedure. This fact, in turn, leads to an important lack of robustness of semiparametric inference procedures, that is, small changes in the choice of tuning and smoothing parameters lead to dramatically different empirical results, making applied work unreliable in general. In other words, this lack of robustness usually translates in incorrect statistical inference that may lead researchers and policymakers to draw flawed conclusions from empirical work that employs these semiparametric inference procedures.The main goal of the proposed research agenda is to develop new, alternative large sample approximations to commonly used semiparametric inference procedures that (at least partially) account for the effect of the specific user-defined choices of tuning and smoothing parameters involved in the inference procedure. This alternative asymptotic theory leads to more "robust" statistical inference procedures because it captures the effect of certain terms that are assumed away by the conventional large sample approximations. This project will proceed in two main stages. First, alternative large sample approximations will be developed for specific semiparametric examples, including weighted averaged derivatives and partially linear model. Not only these models are of interest in their own right, but also they will provide some of the key ingredients to understand the new theoretical features emerging from the non-standard large sample approximations studied in this proposal. Among other problems, the goal is to establish an alternative first-order large sample distribution, derive valid standard-error estimators, develop new ways of selecting the value of the tuning and smoothing parameters, study the validity of commonly used resampling procedures, and explore the higher-order implications of the alternative asymptotic approximations. Once the study of these particular semiparametric procedures is well understood, the second stage of the investigation will be to develop a generalization and unification of the theoretical results outlined for the special examples, which will cover many other problems of interest.The results of this research are expected to benefit several fields of study, ranging from Economics or Political Science to Biostatistics or Public Health, allowing researchers to conduct "robust" inference in semiparametric models, and making semiparametric inference more attractive to researchers and policymakers. To further increase the impact of this research proposal, a key goal is to provide computer code for commonly used platforms, and to write a non-technical survey with a discussion on theory and implementation of both the classical results and the new results emerging from the research proposed.