Abstract
There is growing interest in establishing a mechanism to account for scale heterogeneity across individuals (essentially the variance of a variance term or the standard deviation of utility over different choice situations), in addition to the more commonly identified taste heterogeneity in mixed logit models. A number of authors have recently proposed a model that recognizes the relationship between scale and taste heterogeneity, and investigated the behavioural implications of accounting for scale heterogeneity in contrast to a term in the utility function, itself. In this paper we present a general model that extends the mixed logit model to explicitly account for scale heterogeneity in the presence of preference heterogeneity, and compare it with models that assume only scale heterogeneity (referred to as the scale heterogeneous multinomial logit model) and only preference heterogeneity. Our empirical assessment suggests that accommodating scale heterogeneity in the absence of accounting for preference heterogeneity may be of limited empirical interest, resulting in a statistically inferior model, despite it being an improvement over the standard MNL model. Scale heterogeneity in the presence of preference heterogeneity does garner favour, with the generalized mixed logit model an improvement over the standard mixed logit model. The evidence herein suggests, however, that compared to a failure to account for preference heterogeneity that is consequential, failure to account for scale heterogeneity may not be of such great empirical consequence in respect of behavioural outputs such as direct elasticities and willingness to pay. However additional studies are required to establish the extent to which this evidence is transferable to a body of studies.
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Notes
One can however allow for deterministic taste heterogeneity via interaction terms with respondent-specific characteristics.
Extensive development work was undertaken in the design of the CAPI instrument followed by a pre-pilot of 80 respondents. The pre-pilot data was used to estimate a series of multinomial and nested logit models for the pooled data. On the basis of the review of the pilot output, minor changes to the survey instrument were made.
All models are estimated using (pre-release) Nlogit5.
We have found that using start values from mixed logit for GMXL is preferable than using MNL start values.
AIC = 2 k−2Ln(L) where k is the number of parameters in the model, and L is the maximised value of the likelihood function for the estimated model.
The elasticities are based on uncalibrated models and as such the numerical magnitudes are only valid in the comparisons across models. These models cannot be used to forecast patronage without calibration using revealed preference shares on existing modes.
Since all elasticities are negative, a lower value is an absolute lower value (e.g. −0.435 is lower than −0.650).
We also undertook a bootstrap calculation for two of the variables to ensure that the t-ratio test was a useful approximation. The resulting standard errors confirm that the t-ratios are a good approximation.
In this paper all models are estimated in preference space. We have estimated a GMXL model using the same data in WTP space in Hensher and Greene (2009). We report WTP estimates herein (on the request of referees) although this is not the focus of the paper; however the mean estimates for in-vehicle time, access time and egress time in WTP space for GMXL (M3) given in Hensher and Greene (2009) are respectively $16,35, $24.07 and $15.71 per person hour.
Albeit that problems with it going to infinity are slightly less pronounced than say with the uniform bounded at zero, so that a limited simulation may produce apparently reasonable results.
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The comments of Riccardo Scarpa, John Rose and Chandra Bhat and three referees on earlier versions are appreciated.
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Greene, W.H., Hensher, D.A. Does scale heterogeneity across individuals matter? An empirical assessment of alternative logit models. Transportation 37, 413–428 (2010). https://doi.org/10.1007/s11116-010-9259-z
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DOI: https://doi.org/10.1007/s11116-010-9259-z