Error Components in Demand Estimation
M Vildrik Sørensen, CTT, Technical University of Denmark, DK
Models including error components are today at the frontier of application and development in transport modelling. The name Error Component Models are often used indiscriminately with Random Parameters Logit, Models with Stochastic (or Distributed) Preferences (or Coefficients), Logit Kernel Models or Mixed Logit Models, for models where the error components are added to the traditional (linear) utility function in the following way Ui = (B + K)Xi + E, where B respective E are the preferences respective the unexplained part of the variation, while K is a vector of error components. At present the use of such models is growing rapidly, due to an increased access to. In general the method of Maximum Simulated Likelihood (MSL) is applied, although this only optimises within a given a priori distribution of the error components. Only few of the analyses so far, has dealt with the interesting question of correlation between these error components.
An alternative method to determine the distributions was suggested in Sørensen & Nielsen (2001) and more thoroughly described in Sørensen & Nielsen (2002). The paper uncovers the empirical distribution of the data by repeated estimations. The purpose of the method is to determine the type of distribution, though (for some distributions) it can determine the parameters of the distribution. The authors found, that it is likely that the error components (random coefficients) are lognormally distributed ?and perhaps more interesting, that correlation between the error components is outspoken.
This paper adds on to the paper of Sørensen & Nielsen (2002) with a more comprehensive test of howto incorporate EC?s into traffic models (the construction of the utility function).
A general EC utility function can be written asUi = BXi + F(K) + E ,where B respective E are the preferences respective the unexplained part of the variation, while F(K) isa function of the matrix of error components. The paper compares logit models built, assuming a
* traditional utility function (linear, without EC)
* random coefficients (linear, with EC?s added to the coefficients)
* EC?s on orthogonal elements (principal components), without linear utility (primarily for reference)
* linear utility with EC?s on orthogonal elements (whereby independent distributions of EC can be expected) All utility functions with EC?s are set up two times, specified as independent distributions and simultaneous distributions. This paper makes a full comparison of models based on the above 7 different utility functions (everything else being equal) with a focus on model fit as well as (potentially different) distribution of error components dependent on the functional form of the utility function. The paper will conclude with guidelines on how to include error components in demand estimation.
Association for European Transport