Analysis of Discrete Choice Probit Models with Structured Correlation Matrices

Discrete choice models are very popular in Economics and the conditional logit model is the most widely used model to analyze consumer choice behavior, which was introduced in a seminal paper by McFadden (1974). This model is based on the assumption that the unobserved factors, which determine the consumer choices, are independent and follow a Gumbel distribution, widely known as the Independence of irrelevant Alternatives (IIA) assumption. Alternate models that relax IIA assumption are the Generalized Extreme Value (GEV) models, which allow dependency between unobserved factors. However, GEV models do not incorporate all dependency patterns, other choice behaviors such as random taste variation and repeated responses over time. The discrete choice probit models are the most flexible in the sense that they model any dependence pattern, random taste variations and repeated responses. But, the probit models require evaluations of multivariate normal distribution function, which are difficult to compute. They were not pursued because of this difficulty, except in a few cases with specific patterns in the covariance structures. In this dissertation, we study the discrete choice probit models for a couple of correlation structures such as equicorrelation and product correlation. Using stochastic representations, we derive and simplify analytical expressions for the computation of choice probabilities for both of the structures. Further, we illustrate the procedure of obtaining maximum likelihood estimates for the model parameters and analytical expressions for the Fisher information matrix to compute their standard errors. Using simulations, we compare the performance of probit models with logit models in both large sample case as well as small samples. We conclude that the probit models are more asymptotically efficient than logit models as correlation increases. We have provided a sample R-code in the appendix that was used for computations. Finally, a more general form of choice models are presented using multivariate copulas. We presented a brief introduction of discrete choice copula models using the Gaussian copula and the Extreme value copula. Copula representations are useful in building multivariate distributions with several choices for marginals. The discrete choice probit models are Gaussian Copula models with marginals that are standard normal and the GEV models are Extreme Value Copula models with marginals that are extreme value distributions. This work shows a way of constructing new models using copulas by choosing different marginals within the copula representation. For example, a Gaussian Copula choice model with Gumbel marginals or an Extreme Value Copula choice model with normal marginals is possible. Such models are not yet explored to model consumer choice behavior and this provides a road map for future research