Abstract
My dissertation is composed of four chapters that focus on missing data models, BLP contraction mappings, and Markov chain Monte Carlo estimation. The first chapter focuses on estimating sample selection models with two incidentally truncated outcomes and two corresponding selection mechanisms. The method of estimation is an extension of the Markov chain Monte Carlo (MCMC) sampling algorithm from Chib (2007) and Chib et al. (2009). Contrary to conventional data augmentation strategies for dealing with missing data, the proposed algorithm augments the posterior with only a small subset of the total missing data caused by sample selection. This results in improved convergence of the MCMC chain and decreased storage costs, while maintaining tractability in the sampling densities. The methods are applied to estimate the effects of residential density on vehicle miles traveled and vehicle holdings in California. The empirical results suggest that residential density has a small economic impact on vehicle usage and holdings. In addition, the results show that changes to vehicle holdings from increased residential density are more sensitive for less fuel-efficient vehicles than for fuel-efficient vehicles on average. The second chapter considers the estimation of a multivariate sample selection model with p pairs of selection and outcome variables. A unique feature of this model is that the variables can be discrete or continuous with any parametric distribution, allowing a large class of multivariate models to be accommodated. For example, the model may involve any combination of variables that are continuous, binary, ordered, or censored. Although the joint distribution can be difficult to specify, a multivariate Gaussian copula function is used to link the marginal distributions together and handle the multivariate dependence. The proposed estimation approach relies on the MCMC-based techniques from Lee (2010) and Pitt et al. (2006) and adapts the methods from the preceding authors to a missing data setting. An important aspect of the estimation algorithm, in the same spirit as the algorithm from the first chapter, is that it does not require simulation of the missing outcomes. This has been shown to improve the mixing of the Markov chain. The methods are applied to both simulated and real data. The third paper analyzes a discrete choice model where the observed outcome is not the exact alternative chosen by a decision maker but rather the broad group of alternatives in which the chosen alternative belongs to. This model is designed for situations where the choice behavior at a lower level is of interest but only higher level data are available (e.g. analyzing households’ choices for vehicles at the make-model-trim level but only choice data at the make-model level are observed). I show that the parameters in the proposed model are locally identified, but for certain configurations of the data, they are weakly identified. Methods to incorporate additional information into the problem are discussed, and both maximum likelihood and Bayesian estimation methods are explored. The last chapter proposes improvements to the contraction mappings used in the context of multinomial logit models. The contraction mapping algorithm proposed in Berry et al. (1995) is slow to converge and is a major burden to implement in applied work. While it is relatively quick to converge for a single run of the algorithm, it is computationally expensive when repeated evaluations are needed, particularly when the algorithm is embedded into maximum likelihood, generalized method of moments, or Bayesian Markov chain Monte Carlo estimation routines. To alleviate this problem, I explore four simple modifications of the contraction mapping to improve its rate of convergence. Importantly, the modifications can be incorporated into existing code with minimal effort. In a simulation study, I demonstrate that the new algorithms require significantly fewer iterations to converge to the unique vector of fixed points than the original specification. The best algorithm results in an 80-fold improvement.