On the equivalence of common rearrangements for sum constrained travel estimation problems

This paper examines the problem of estimating the parameters of nonlinear sum constrained models by use of rearrangements to permit linear estimation methods. Over the years a number of different, and sometimes ingenious, techniques have been proposed but a remaining difficulty arises from the fact that different rearrangements lead to different parameter estimates. In this paper it is shown that while ordinary least squares certainly results in different estimates, generalised least squares leads to identical parameter estimates provided convergence to the true maximum likelihood estimates is achieved. In particular it is seen that estimates are invariant with choice of base or deletion of one equation and identical for the single base, multinominal logit, geometric mean and link constant versions. The findings of this paper apply to any sum-constrained model and in transportation these occur very frequently as mode choice problems, singly-constrained gravity models, and as route choice models in path assignment. While we develop all results in this paper in the context of an intercity passenger modal market share model, the transferability of these results to other problems should be recognised.