A Tour-Based Model of Travel Mode Choice - Civil Engineering ...

10 th International Conference on Travel Behaviour Research
August 10-15, 2003
Ortúzar, J. de D. (1983) Nested Logit Models for Mixed-mode Travel in Urban Corridors,
Transportation Research 17A (4), 283-299.
Ortúzar, J. de D. and L. Willumsen (2001) Modelling Transport, Chichester NY: Wiley.
Train, K. (2002) Discrete Choice Methods with Simulation, Cambridge: Cambridge
University Press, preprint available at http://else.berkeley.edu/books/train1201.pdf.
Appendix A: Statistical Identification of Model Parameters
The procedure followed here is inspired by Bunch (1991) and Ben-Akiva, et al., (2001). It
involves examining whether the model meets order, rank and positive definiteness conditions.
Recall that the model error structure is:
ε(m,t,p) = µ(m,p) + η(m,t,p)
[A.1]
where:
µ(m,p) ≈ MVN(0,Σ)
η(m,t,p)≈ N(0,σ)
Two cases are considered:
• a homoscedastic iid model, which has two parameters: one along the diagonal of the
matrix Σ, and the term σ; and
• a heteroscedastic, independent model, which has M+1 parameters: M diagonal elements
in the matrix Σ (where M is the number of trip modes in the model) and the
term σ.
The order condition states that a maximum of M(M-1)/2 alternative-specific parameters are
estimable in Σ. In this case, M = 3; thus, an upper bound on the number of estimable parameters
in Σ is 3.
The rank of Σ depends on its specification. In the iid case, the rank of the matrix is 1; in the
heteroscedastic case, the rank is 3. The “rank” of σ, of course, is 1 for both models.
The positive definiteness condition is not binding for this model due to its similarities with a
multinomial probit model, for which any positive normalization is acceptable (Ben-Akiva, et
al., 2001).
Finally, the overall utility function U(m,t,p) (equation [1]) is only identified up to scale, resulting
in a loss of one estimable parameter. Thus, the number of identifiable parameters in
the model is the sum of the ranks of the two error terms, minus one, to account for the utility
23