Collaborative Research: A Markov Chain Approach to Classical Estimation

这个项目开发了一类新的估计量的统计性质，称为拟贝叶斯(QBE)或拉普拉斯估计量。这些估计量适用于高度非线性的经典M估计问题，以及许多非光滑半参数估计问题。他们利用贝叶斯统计中蒙特卡罗马尔可夫链模拟的最新发展，克服了许多非线性经典估计器的计算困难。这一研究项目的结果有助于理解一般非线性计量经济模型中的计算和推断。计算和推断是任何计量经济学模型不可分割的两个重要部分。建议的估计器解决了这两个问题，并有可能在实证工作中广泛使用。许多参数和半参数估计涉及到非凸和非光滑的目标函数。这不仅使大样本统计性质的证明变得困难，而且如果可能的话，也使估计量在实践中难以计算。本项目的第一部分定义了旨在克服这两个突出问题的拟贝叶斯估计，并研究了它们的正则相合性和渐近正态性质。当基本目标函数是数据的实际对数似然函数时，QBE简化为通常的贝叶斯方法。第二部分研究了一类非正则结构计量经济模型中基于似然的推断的性质，包括贝叶斯估计和极大似然估计，其中因变量的支持度既可以依赖于模型参数，也可以依赖于自变量。这些模型是在结构性拍卖模型、经验均衡求职模型和前沿生产分析的背景下自然产生的。该项目的这一部分提供了在这些模型中基于似然估计和推断的统一处理。前两部分适用于有限维参数。第三部分将拟贝叶斯方法推广到同时估计有限和无限维参数的方法，其中无限维参数用筛子空间来逼近。拟贝叶斯估计克服了筛分半参数模型的计算障碍。第四个项目分析了参数在参数空间边界上时的QBE的性质，这种性质可以通过对参数的线性或非线性约束来定义。虽然文献中对M估计量的性质进行了广泛的分析，但当参数位于参数空间的内部时，它们并不具有M估计量的最优性。QBE提供了有用的替代最优估计器，并具有易于处理的计算和推理特性。第五部分重点讨论了QBE的计算性质和一系列具体问题的实现细节。研究了用于监控QBE计算的收敛准则，以及对流行的遗传优化算法模拟退火法的改进，用其他Gibbs抽样算法取代了遗传Metropolis步长。

This project develops the statistical properties of a class of new estimators, called quasi-Bayesian (QBE) or Laplacian estimators. These estimators are applicable to highly nonlinear classical M estimation problems, and to many non-smooth semiparametric estimation problems. They use the latest development in Monte Carlo Markov Chain simulations in Bayesian statistics to overcome the computational difficulty of many nonlinear classical estimators. The results of this research project contribute to the understanding of computation and inference in general nonlinear econometric models. Computation and inference are two inseparable essential parts of any econometric model. The proposed estimators address both issues and can potentially be used extensively in empirical work. Many parametric and semiparametric estimators involve non-convex and nonsmooth objective functions. This not only makes proving large sample statistical properties difficult, but also makes the estimators difficult, if possible, to compute in practice. The first part of this project defines quasi-Bayesian estimators that aim at overcoming these two prominent issues, and studies their regular consistency and asymptotic normality properties. When the underlying objective function is an actual log likelihood function of the data, the QBEs reduce to the usual Bayesian approach. The second part studies the properties of likelihood based inferences, including Bayesian and maximum likelihood estimators, in a class of nonregular structural econometric models, in which the support of the dependent variable can depend on both the model parameters and the independent covariates. These models arise naturally in the context of structural auction models, empirical equilibrium job search models and frontier production analysis. This part of the project provides a unified treatment of likelihood based estimation and inference in these models. The first two parts apply to finite dimensional parameters. The third part extends the quasi-Bayesian approach to estimate finite and infinite dimensional parameters simultaneously, in which the infinite dimensional parameter is approximated by a sieve space. The quasi-Bayesian estimator overcomes the computational obstacle of sieve semiparametric models. The fourth project analyzes the properties of QBEs when the parameters are on the boundary of the parameter space, which can be defined through linear or nonlinear constraints on the parameters. While the properties of M estimators have been extensively analyzed in the literature, they do not share the optimality properties of M estimators when the parameter is in the interior of the parameter space. QBEs provide useful alternative optimal estimators, and enjoy tractable computational and inference properties. The fifth part focuses on the computational properties of QBEs and the implementation details for a range of specific problems. It studies convergence criteria that are used to monitor the computation of QBEs and improvements to the popular generic optimization algorithm simulated annealing, by replacing the generic Metropolis step with other Gibbs sampling algorithms.