Abstract
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.
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Notes
Batley (2008) notes that McFadden’s work actually dates back to 1968, when it first appeared as an unpublished working paper at the University of California, Berkeley.
This same result was found for the two log-linear direct utility functions considered in the Train and McFadden (1978) paper.
Train and McFadden (1978) derive Eqs. (6b) and (6c) from the following direct utility functions: \( U = {\alpha_1}\log G + {\alpha_2}L \) and \( U = {\alpha_1}G + {\alpha_2}\log L \). In maximazing them, respectively α2L and α1G drop out, hence the indirect utilities (6b) and (6c) are obtained, where actually α 1 = 1-β and α 2 = β.
Note that neither the wage rate nor the direct preference parameter β depends on the discrete alternative. Hence, although β plays a relevant role on the goods/leisure trade-off, under a strictly deterministic approach it has no influence on the discrete choice. See Cherchi (2003) and Amador (2005) for more details.
They employ a similar reasoning to justify what they called the expenditure rate specification. Nonetheless, some years later Jara-Díaz (1991, p.343) showed that the aforementioned specification was actually a particular case of the generalized expenditure rate model. On the other hand, Train and McFadden (1978, p.350) suggest two alternative ways of deriving the CIUF as equivalent approaches and they explicitly recognize that the CIUF obtained through one manner is the same or a monotonic transformation of the CIUF resulting from the other approach. Thus, they are implicitly accepting the use of order-preserving transformations of the CIUF.
Note that we use the term direct preference parameter to denote the parameter β influencing the direct utility function. This parameter should not be confused with the parameters θ that appear into the representative utility function, which also reflects individuals’ preferences.
The generalization proposed by Train and McFadden (1978) postulates also the existence of different components of travel time and cost: \( {U_j} = - k\sum\nolimits_{i = 1,..,N} {\delta_c^i{\omega^{ - \beta }}c_j^i} - k\sum\nolimits_{i = 1, \ldots M} {\delta_t^i{\omega^{1 - \beta }}t_j^i} - k\left( {{\eta_{cj}} + {\eta_{tj}}} \right) \). However, considering only one component of time and cost does not affect our discussion.
In fact, this argument could partially explain empirical findings by Amador et al. (2008) on confounding preference heterogeneity and income effect.
This is an assumption that might be discussed, but it was based on average real data, in order to generate a fairly realistic choice set. Following Train and McFadden (1978) the endowment represents the income not generated by work.
These are the correct specifications because are the specifications used to generate the data.
We generated both the ER and WR data using the same process, trying to keep the two dataset as similar as possible in terms of values of the variables involved. However because of differences in the constraints, some explicative variables and of course the final choices are different. So the LL(WR) and LL(ER), even in the true specification, cannot actually be compared among them.
Note that a linear in time and cost specification like WR1 is not an order preserving transformation of the true non-transformed utility (4) even from a deterministic perspective, hence it was expected that models WR2 and WR3 fitted the data better than model WR1.
A normal distribution is assumed for the random parameters in all the Mixed Logit models. Models WR12, WR25, ER8-ER10, assume independent random parameters, while the remaining Mixed Logit models account for correlation between random parameters.
Two income strata were considered; therefore two dummy variables were defined: LowInc = 1 if total income is less or equal than 75 euros while MidInc = 1 if total income is between 75 and 125 euros. The total income is given by the sum of endowment and wage income (E+ωW).
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Acknowledgements
This research was partially done during a stay of the second author at La Laguna funded by the Universidad de La Laguna Programme for Research Support. We would also like to thanks Juan de Dios Ortúzar for his amazing job in improving the clarity of the paper.
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Amador, F.J., Cherchi, E. Econometric Effects of Utility Order-Preserving Transformations in Discrete Choice Models. Netw Spat Econ 11, 419–438 (2011). https://doi.org/10.1007/s11067-010-9134-7
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DOI: https://doi.org/10.1007/s11067-010-9134-7