Econometric Effects of Utility Order-Preserving Transformations in Discrete Choice Models

In the random utility modelling context, choice probabilities are unaffected by increasing linear transformations of the systematic utility; hence its empirical specification is derived on the basis that only differences in utility matters and that the scale of utility is arbitrary. We argue that choice probabilities remain unchanged if these linear transformations are made under the deterministic perspective of a single individual choosing several times. But, in the random utility setting, parameter estimates might be significantly affected by these transformations. In particular we focus on the effect of two order-preserving transformations usually applied in the derivation of the representative utility from the conditional indirect utility function: adding a constant to the utility of all alternatives and multiplying each alternative utility by a constant. We concentrate on the two most popular specifications in transport mode choice: the "wage rate" (Train and McFadden Transport Res 12:349-353, 1978) and the "expenditure rate" (Jara-Díaz and Farah Transport Res 22B:159-171, 1987) specifications. Using a collection of synthetic datasets generated in a new fashion directly from the conditional indirect utility function, i. e. before applying any expansion or transformation, we demonstrate how taking this class of order-preserving transformations could lead to misinterpretation of the econometric results, such as detecting randomly distributed and correlated parameters and/or income and time effects which are in fact not present.