Topics in econometrics and optimization
- This thesis investigates the interesting connections and interactions between some classic topics in econometrics and optimization. We show that insights and results from one area can help solve hard problems in the other, and that there is value in connecting seemingly unrelated disciplines and problems. We will focus on two important models of discrete choice: probit and logit models. Probit models are important tools for modeling discrete responses economics, finance, statistics, and epidemiology. However, the latent nature of probit coupled with identification constraints pose significant computational challenges for its estimation, especially when the dimension of the response variable is large. We propose a computationally efficient EM algorithm for estimating large multi-dimensional probit models. We use expectation propagation (EP) instead of simulation methods to approximate moments of the truncated multivariate normal (TMVN) in the E (expectation) step. In addition, we transform the constrained optimization problem in the M (maximization) step into a one-dimensional problem, which is solved directly using Newton's algorithm. Our method enables the analysis of choice data in the presence of many alternatives and features common in modern applications but has been difficult in practice with probit models. In the second part of the thesis, we show, for a broad class of choice and ranking models based on Luce's choice axiom, that the associated maximum likelihood estimation problem is equivalent to a classic matrix balancing problem with target row and column sums. We draw inspirations from these connections and resolve important open problems on the study of Sinkhorn's algorithm. We first prove the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite solutions to the balancing problem exist. We characterize this global rate of convergence in terms of the algebraic connectivity of the bipartite graph constructed from data. Next, we also derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008). To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals.
|Type of resource
|electronic resource; remote; computer; online resource
|1 online resource.
|Degree committee member
|Degree committee member
|Stanford University, School of Humanities and Sciences
|Stanford University, Department of Economics
|Statement of responsibility
|Submitted to the Department of Economics.
|Thesis Ph.D. Stanford University 2023.
- © 2023 by Zhaonan Qu
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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