Get me out of this hole: a profile likelihood approach to identifying and avoiding inferior local optima in choice models

Abstract: Choice modellers routinely acknowledge the risk of convergence to inferior local optima when using structures other than a simple linear-in-parameters logit model. At the same time, there is no consensus on appropriate mechanisms for addressing this issue. Most analysts seem to ignore the problem, while others try a set of different starting values, or put their faith in what they believe to be more robust estimation approaches. This paper puts forward the use of a profile likelihood approach that systematically analyses the parameter space around an initial maximum likelihood estimate and tests for the existence of better local optima in that space. We extend this to an iterative algorithm which then progressively searches for the best local optimum under given settings for the algorithm. Using a well known stated choice dataset, we show how the approach identifies better local optima for both latent class and mixed logit, with the potential for substantially different policy implications. In the case studies we conduct, an added benefit of the approach is that the new solutions exhibit properties that more closely adhere to the property of asymptotic normality, also highlighting the benefits of the approach in analysing the statistical properties of a solution.