Efficient Resampling and Simulation Methods for Nonlinear Econometric Models

经济学和其他学科的许多统计估计和推理程序依赖于密集的计算，其中包括使用计算机模拟和重新采样方法，这些方法重用用于经济分析的随机数据集。该方案主要包括两个项目，分别研究基于模拟和重采样的估计器的统计特性，这两个项目适用于高度非线性和计算密集型模型。这一点很重要，因为忽略模拟和重采样方法引入的统计不确定性可能导致统计和经济分析中的错误结论。产生随机数很容易，但计算矩条件或似然函数通常很困难。对所有观测数据使用重叠模拟是否能提高计算效率取决于具体的模型。第一个项目的目标是研究一种特定类型的基于模拟的估计程序的大样本分布，其中所有观测都使用相同的模拟图集。这里考虑两个重要的案例。这包括求解模拟矩（MSM）系统的估计器和最大化模拟似然（MSL）的估计器。在这个项目中发展的理论适用于经验工作中使用的许多模拟估计器，这些模拟估计器涉及重叠模拟绘制和不可微矩函数。证明了在样本量和模拟次数无限制增加的情况下，MSM和MSL是一致的。在适当的规则性条件下，MSM和MSL都以观测数和模拟图数最小值的平方根的速率收敛到一个极限正态分布。MSM和MSL的条件不同。对于MSL，一方面，模拟次数的增长速度必须快于样本量的平方根，这是具有独立随机抽取的渐近正态性所需要的条件。另一方面，对于重叠绘制，只要模拟次数和观测次数都增加到无穷大，渐近正态性就保持不变。我们还发现，模拟的总数必须无限制地增加，但可以比观测的总数小得多。在这种情况下，参数估计误差主要由仿真误差控制。当仿真模型需要大量计算时，这是必要的推理代价。第二个项目提出了一种快速采样方法，可以用于非线性参数和半参数模型的有效推理。该方法不需要在每次重抽样迭代中重新计算第二阶段估计量，但仍然在非常弱的假设下为大类非线性模型提供了有效的推断。这些模型在需要估计的参数上可以是高度非线性的，也可以是半参数的，通过依赖于第一阶段的非参数函数估计过程。快速重抽样方法直接利用每个bootstrap样本上计算的分数函数表示，从而大大减少了计算时间。本文提出的方法也可以扩展到第一阶段计算比第二阶段计算更密集的模型中，通过在第二阶段估计过程重采样时使用第一阶段的线性表示。在蒙特卡罗实验中证明了快速自举法的良好性能和在数值速度上的巨大改进。在第一个项目中发展具有重叠绘图和非光滑函数的抽样定理，为模拟估计量渐近性的现有文献结果提供了重要的补充。第二个方案中的快速重采样方法用于逼近参数估计量和半参数估计量的极限分布，这些估计量可能是基于仿真的，允许渐近线性表示。它还可以用于减少偏差和方差估计，这是经验模型的计量经济推断的重要组成部分。从项目中获得的结果可以为大量使用计算密集型非线性模型的经验研究人员提供非常有用的指导，对于这些模型，获得估计量和对感兴趣的参数进行推理都是具有数值挑战性的。除了在经济学中的应用之外，非线性模型也广泛应用于统计学以及社会科学和自然科学的各个学科，研究人员经常采用基于模拟和重抽样的方法进行估计和推断。这种分析可以为经验研究人员提供指导，通过阐明对模拟和重采样过程引入的统计不确定性的理解和核算，利用这些模型。

Many procedures of statistical estimation and inference in economics and in other disciplines rely on intensive computation, which include the use of computer simulations and resampling methods that reuse a random set of the data that is being used for economic analysis. This proposal consists mainly of two projects that study the statistical properties of simulation and resampling based estimators, which are applicable to highly nonlinear and computationally intensive models. This is important because ignoring the statistical uncertainty introduced by simulation and resampling methods can lead to erroneous conclusion in the statistical and economic analysis. Generating the random numbers is easy but computing the moment condition or the likelihood function is typically difficult. Whether using overlapping simulations for all observations presents an improvement in computational efficiency depends on the specific model.The goal of the first project is to study the large sample distribution of a particular type of simulation based estimator procedures where the same set of simulation draws are used for all observations.Two important cases are considered. These include estimators that solve a system of simulated moments (MSM) and estimators that maximize a simulated likelihood (MSL). The theory being developed in this project applies to many simulation estimators used in empirical work which involve both overlapping simulation draws and non-differentiable moment functions. It is proven that both MSM and MSL are consistent when both the sample size and the number of simulations increase without bound. Under suitable regularity conditions, both MSM and MSL converge at the rate of the square root of the minimum of the number of observations and the number of simulation draws, to a limiting normal distribution.The conditions differ between MSM and MSL. For MSL, on the one hand, the condition that the number of simulations has to increase faster than the square root of the sample size is needed for asymptotic normality with independent random draws. On the other hand, with overlapping draws, asymptotic normality holds as long as both the number of simulations and the number of observations increase to infinity. It is also found that the total number of simulations has to increase without bound but can be much smaller than the total number of observations. In this case, the error in the parameter estimates is dominated by the simulation errors. This is a necessary cost of inference when the simulation model is very intensive to compute.The second project proposes a fast resample method that can be used to provide valid inference in nonlinear parametric and semiparametric models. This method does not require recomputation of the second stage estimator during each resample iteration but still provides valid inference under very weak assumptions for a large class of nonlinear models. These models can be highly nonlinear in the parameters that need to be estimated and can also be semiparametric through dependence on a first stage nonparametric functional estimation procedure. The fast resample method directly exploits the score function representations computed on each bootstrap sample, thereby reducing computational time considerably. The method presented here can also be extended to models in which the first stage computation is more intensive than the second stage, by making use of a linear representation for the first stage when resampling the second stage estimation procedure. The desirable performance and vast improvement in the numerical speed of the fast bootstrap method are demonstrated in the Monte Carlo experiments that have thus far been conducted.Developing sampling theorems with overlapping draws and nonsmooth functions in the first project provides an important complement to the existing results in the literature on the asymptotics of simulation estimators. The fast resampling method in the second project is used to approximate the limit distribution of parametric and semiparametric estimators, possibly simulation based, that admit an asymptotic linear representation. It can also be used for bias reduction and variance estimation, which are important components for the econometric inference of empirical models. The results obtained from the project can provide very useful guidance to empirical researchers who make extensive use of computational intensive nonlinear models for which obtaining the estimator and conducting inference on the parameter of interest can both be numerically challenging. Beyond applications in economics, nonlinear models are also widely used in statistics and various disciplines in social sciences and natural sciences, where researchers often resort to simulation and resampling based methods for estimation and inference. This analysis can provide guidance to empirical researchers making use of these models by shedding light on understanding and accounting for the statistical uncertainty introduced by the simulation and resampling procedures.