Linear probit models: Statistical properties and improved estimation methods

The multinomial probit model is a statistical tool that is well suited to analyze some transportation problems. Modal split, gap acceptance, and route choice are some examples of application contexts. This paper presents an in-depth analysis of its statistical properties and an estimation method for the trinomial case. In the statistical part of the paper it is shown that for multinomial probit models with specifications that are linear in the parameters, the global maximum of the log-likelihood function is consistent if the data do not exhibit multicollinearity as defined in the text. For the special case with three alternatives, lack of multicollinearity is also shown to guarantee asymptotic efficiency and normality, and the uniqueness of any root of the likelihood equations. In addition, it is also shown that for the trinomial probit model certain goodness-of-it measures and test statistics can be easily calculated. The methods part of the paper introduces an estimation process that solves the likelihood equations using a special purpose table of the bivariate normal distribution and analytical derivatives of the log-likelihood function. The method is very accurate, can be applied to nonlinear specifications, and is considerably faster than current computer programs. For linear specifications, the method can be mathematically proven to converge if the log-likelihood equations have a root.