A new mixed MNP model accommodating a variety of dependent non-normal coefficient distributions

An analogous structure may be obtained by essentially adding an IID Gumbel error term across alternatives to the multivariate normal coefficients, leading to a mixed multinomial logit model; see Bhat (1997) and Revelt and Train (1998) for the first multivariate applications of this type of a model. Alternatively, one can add a multivariate extreme value (MEV) error vector kernel to the utility of the alternatives, combined with additional non-identical kernel error terms, to the random coefficients vector (see, for example, Bhat and Guo 2007). But, as discussed in detail in Bhat (2011), all these structures essentially achieve the same purpose, and the choice is simply a matter of convenience. Besides, the use of an MNP kernel has substantial advantages when combined with recently proposed analytic methods of evaluating a multivariate cumulative normal distribution (MVNCD) function that have been shown to be much more computationally efficient than traditional simulation approaches. Also, when extensions to accommodate correlation across decision makers due to spatial and/or social interactions are considered, the MNP kernel is much easier and more efficient. We will henceforth focus in this paper on the MNP kernel.

Just to clarify a myth. The mixed multinomial logit model is no more general than the mixed MNP model, as long as we allow the mixing distribution with the MNP kernel to be non-normal, as we do so in the current paper.

Discrete distributions may also be used for the mixing. If the mixing vector is assumed to take M possible value states with state-specific probabilities, this leads to the familiar latent class model used in marketing (see Kamakura and Russell 1989) and transportation (see Bhat 1997). On the other hand, if a discrete distribution is considered separately for each individual random coefficient, this is essentially a non-parametric random coefficients model (see Bastin et al. 2010; Berry and Haile 2014; il Kim 2014). The non-parametric specification allows consistent estimates of the observed variable effects under broad model contexts by making regularity (for instance, differentiability) assumptions on an otherwise distribution-free density form. But the flexibility of these methods comes at a high inferential cost since consistency is achieved only in very large samples, parameter estimates have high variance, and the computational complexity/effort can be substantial (Mittelhammer and Judge 2011). Overall, the continuous distribution specification dominates the literature, at least in part because it offers efficiency in the number of mixing distribution parameters to be estimated.

Note that, by construction, the marginal multivariate distribution function of \({\varvec{\upbeta }}_{q}\) is the multivariate standard normal distribution function of \({\tilde{\varvec{\upbeta }}}_q \); that is \(F_E ({\varvec{\upbeta } }_q <\mathbf{z}_q )=\Phi _E ({{\varvec{g}}}_q \hbox {;} {\varvec{\Gamma }}_{\tilde{\beta }} ),\) from which \(f_E (\mathbf z _q )=\frac{dF_E ({\varvec{\upbeta } }_q <\mathbf z _q )}{{\varvec{dz}}_q }= \phi _E ({{\varvec{g}}}_q \hbox {;} {\varvec{\Gamma }}_{\tilde{\beta }} )\frac{{\varvec{dg}}_q }{{\varvec{dz}}_q },\) or \(f_E (\mathbf z _q ){\varvec{dz}}_q =\phi _E ({{\varvec{g}}}_q \hbox {;} {\varvec{\Gamma }}_{\tilde{\beta }} ){{\varvec{dg}}}_q \), and Eq. (14) is the result.

On the other hand, the problem with the log-normal distribution to represent a coefficient such as a cost coefficient is that the tails of the distribution are directly determined by the variance term. If there is high heterogeneity in the sensitivity to cost, this immediately implies a peaking (mode) close to zero as well as a long and fat left tail (note that the cost coefficient is introduced as the negative of the log-normal distribution). The result is that, as the variance parameter of the log-normal distribution increases (for the same mean parameter), a larger fraction of individuals will have a small cost coefficient. At the same time, a small fraction of individuals will have very high cost sensitivity because of the long and fat tail. The result can cause unusually large and small willingness to pay estimates. Further, the long and fat tail on the unbounded side of the distribution is known to cause convergence problems during estimation (Bartels et al. 2006).

As discussed earlier, the log-normal distribution a priori fixes the power term to 1. Here, while we can estimate the power term, our experience suggested that the optimization algorithms took longer with much more convergence difficulty than if the power term was fixed. That is, the best way to estimate a model with a power log-normal term appears to be to estimate the model at different fixed values of the power term, and then compare the data fits across the different optimization function values (corresponding to different fixed values of the power term) to determine the best value for the power term. That is the reason we fix the power term at the value of five in the simulation estimations here, while estimating the means (\(\mu _{1}\) and \(\mu _{2}\)) and the scale parameters ( \(\sigma _{1}\) and \(\sigma _{2}\)).

The specification for the differenced covariance matrix above may be viewed as being derived from a specification where the error terms for the first three alternatives are independent and distributed with a variance of 0.5, while the last error term has a variance of 0.913 and is correlated with the error term of the third alternative with a covariance of 0.1. In the simulation experiment estimations, to focus on the random coefficients, we fix the variances of the first three alternatives to 0.5 and impose independence among the first three alternatives, but estimate the variance of the fourth error term and the covariance between the third and fourth alternatives, which translates to the two Cholesky parameters \(l_{\varTheta 5} =0.404 \hbox { and } l_{\varTheta 6} =0.998.\)

This penalized log-composite likelihood is nothing but the generalization of the usual Akaike’s Information Criterion (AIC). In fact, when the candidate model includes the true model in the usual maximum likelihood inference procedure, the information identity holds (i.e., \({{\varvec{H}}} ({\varvec{\theta }})={{\varvec{J}}}({\varvec{\theta }}))\) and the CLIC in this case is exactly the AIC \([=\log L_{ML} (\hat{{{\varvec{\theta }} }})-\)(# of model parameters)].

There has been a healthy discussion and debate in the literature (see, for example, Ory and Mokhtarian 2005; Cirillo and Axhausen 2006) on the issue of whether or not some individuals associate a positive valuation to travel time as opposed to the predominantly held view that people are averse to higher travel times. Of course, there is also the issue that this may be very context dependent, including, for example, the length of the travel time being considered (see, for example, Pinjari and Bhat (2006), who suggest that the sensitivity to travel time is non-linear over travel time). In this paper, we do not engage in this line of debate. The purpose here is to present, and demonstrate an application of, a flexible copula model and its estimation that can be gainfully employed to estimate different combinations of multivariate random coefficient distributions to then guide the final model structure and specification, based on theoretical considerations (for example, which coefficients should have bounded distributions and which can have unbounded distributions), intuitive considerations (the reasonableness of trade-off values obtained and their profiles over the population), and statistical data fit considerations.