Testing for regularity and stochastic transitivity using the structural parameter of nested logit

Abstract

We introduce regularity and stochastic transitivity as necessary and well-behaved conditions respectively, for the consistency of discrete choice preferences with the Random Utility Model (RUM). For the specific case of a three-alternative nested logit (NL) model, we synthesise these conditions in the form of a simple two-part test, and reconcile this test with the conventional zero-one bounds on the structural (‘log sum’) parameter within this model, i.e. 0 < θ ≤ 1, where θ denotes the structural parameter. We show that, whilst regularity supports the lower bound of zero, moderate and strong stochastic transitivity may, for some preference orderings, give rise to a lower bound greater than zero, i.e. impose a constraint l ≤ θ, where l > 0. On the other hand, we show that neither regularity nor stochastic transitivity constrain the upper bound at one. Therefore, if the conventional zero-one bounds are imposed in model estimation, preferences which violate regularity and/or stochastic transitivity may either go undetected (if the ‘true’ structural parameter is less than zero) and/or be unknowingly admitted (if the ‘true’ lower bound is greater than zero), and preferences which comply with regularity and stochastic transitivity may be excluded (if the ‘true’ upper bound is greater than one). Against this background, we show that imposition of the zero-one bounds may compromise model fit, inferences of willingness-to-pay, and forecasts of choice behaviour. Finally, we show that where the ‘true’ structural parameter is negative (thereby violating RUM – at least when choosing the ‘best’ alternative), positive starting values for the structural parameter in estimation may prevent the exposure of regularity and stochastic transitivity failures.