Collaborative Research: A Markov Chain Approach to Classical Estimation

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

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.