Date of Award
Spring 2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Economics
First Advisor
Chen, Xiaohong
Abstract
A substantial body of research focuses on the statistical analysis of economic models featuring either endogenous regressors or latent unobservable variables. These models are often characterized by nonlinear and possibly nonsmooth objective functions, low signal-to-noise ratios and high sensitivity to user-chosen tuning parameters. This thesis comprises three chapters, each proposing and developing novel methodology for nonparametrically estimating a broad class of economic models characterized by these features. In the first chapter, "Adaptive Estimation and Inference in Nonparametric IV Models" (with Xiaohong Chen and Tim Christensen), we introduce two data-driven procedures for optimal estimation and inference in nonparametric models using instrumental variables. The first is a data-driven choice of sieve dimension for a popular class of sieve two-stage least squares estimators. When implemented with this choice, estimators of both the structural function and its derivatives converge at the fastest possible (minimax) rate in supremum risk. The second is for constructing uniform confidence bands (UCBs) for the structural function and its derivatives. Our UCBs guarantee coverage over a generic class of data-generating processes and contract at the minimax rate, possibly up to a logarithmic factor. As such, our UCBs are asymptotically more efficient than UCBs based on the usual approach of undersmoothing. As an application, we estimate the elasticity of the intensive margin of firm exports in a monopolistic competition model of international trade. The second chapter, ``Quasi-Bayes in Conditional Moment Restriction Models'' studies quasi-Bayesian estimation and uncertainty quantification for an unknown structural function that is identified by a conditional moment restriction. Many economic models, such as nonparametric quantile instrumental variables, are typically identified through restrictions which induce nonsmooth, nonlinear and ill-posed objective functions. A quasi-Bayes approach views this objective function as a pseudo-likelihood for the model. When combined with a prior, this provides a researcher with quasi-Bayes decision rules such as point estimators (e.g. posterior means) and credible sets. Among other factors, quasi-Bayesian approaches are particularly attractive in such setups as they avoid numerical optimization of the objective function and implicitly regularize the problem through a prior. I derive nonparametric contraction rates for a class of Gaussian process priors. Furthermore, I provide conditions under which a Bernstein–von Mises theorem holds for the quasi-posterior distribution. As a consequence, I show that optimally weighted quasi-Bayes credible sets have exact asymptotic frequentist coverage. This extends classical results on the frequentist validity of optimally weighted quasi-Bayes credible sets for parametric generalized method of moments (GMM) models. The third chapter, ``Quasi-Bayes in Latent Variable Models'' extends the investigation of quasi-Bayesian procedures to economic models with latent unobservables. In these settings, the unknown parameter is an infinite dimensional latent distribution of interest. My analysis begins with the observation that a large class of latent variable models are characterized by a collection of simple identifying restrictions that relate the characteristic function of the observables to the distribution of the unobservables. I use these restrictions to build a pseudo-likelihood for the model. This pseudo-likelihood is then combined with a prior to provide a quasi-Bayes posterior. I propose a class of priors that are supported on Gaussian mixtures. In this setting, a desirable property of focusing on Gaussian mixture based priors is that priors on Gaussian mixtures translate effortlessly to priors on characteristic functions and vice versa. I develop the quasi-Bayes limit theory for three classes of latent variable models: models with classical measurement error, models with repeated measurements and linear multi-factor models. In each case, I derive rates of convergence (posterior contraction rates) for the induced quasi-Bayes posterior. As my analysis is the first quasi-Bayesian approach to these classes of models, I expect that the general analysis may be of independent interest towards related extensions. In a simulation study, I illustrate the favorable performance of quasi-Bayes estimators relative to all existing alternatives. As a first application, I use data from the India Young Lives Survey to estimate production functions for cognition and health for children aged 1-12 in India. As a secondary application, I model individual log earnings from the Panel Study of Income Dynamics (PSID) as the sum of permanent and transitory components.
Recommended Citation
Kankanala, Sid, "Essays in Nonparametric Econometrics: Endogeneity and Latent Heterogeneity" (2024). Yale Graduate School of Arts and Sciences Dissertations. 1285.
https://elischolar.library.yale.edu/gsas_dissertations/1285
