Estimation of consistent Logit and Probit models using best, worst and best–worst choices

Abstract

The paper considers random utility models that use a single common vector of random utilities for the computation of best, worst and best–worst choice probabilities, i.e. consistent models. Choice probabilities are derived for two distributions of the random terms: i.i.d. extreme value, i.e. Logit, and multivariate normal, i.e. Probit. We prove strict log-concavity of the likelihood, with respect to the coefficients of the systematic utilities, for best, worst and best–worst choice probabilities in Logit, and for best and worst choice probabilities in Probit, under a mild necessary and sufficient condition of absence of perfect multicollinearity in the matrix of alternative and individual characteristics. This condition parallels that in ordinary least squares linear regression models. The hypothesis of equality of the utility coefficients of best choice models and of worst choice models is tested with data on mode choice, collected for the assessment of user responses to urban congestion charging policies. The numerical results show, in both Logit and Probit, statistically significant differences between utility coefficients of best and worst models. The estimations based on worst choice data exhibit coefficient attenuation and higher mean values of travel time savings with larger standard errors.