Elasticity, Model Scale and Error
A Daly, RAND Europe, UK
An explanation of apparent paradoxes in the interaction of model sensitivity (elasticity), model scale and error
Elasticity is often used as a convenient means of summarising the sensitivity of a population to changes in transport policy variables, such as the prices of alternative modes or routes. In developing models analysts are often required to demonstrate that the elasticities implied by their models are consistent with generally accepted notions of what elasticities ought to be. Most travel demand models used in practical forecasting studies are formulated with a single measure of utility (or its negative, generalised cost) for each alternative mode, route (or destination etc.) and in such models it is clear that the elasticity they yield is directly proportional to the scale that multiplies the utility of the alternatives. Obtaining an acceptable value for elasticity is then a matter of obtaining a suitable value for that scale, conditional on the structure of the model.
However, it is generally known that when statistical estimates are made of the coefficients of a model the value of the model scale is inversely proportional to the error with which utility is measured. There is therefore an apparent paradox that an improvement to the model, i.e. a reduction in error, hence an increase in the model scale parameter, seems to indicate an increase in elasticity. It cannot be the case that the quality of the model affects the behaviour of the population!
The paper presents a resolution of this apparent paradox, which is analysed in terms of the distribution of the utility function among the population. When a model is improved, although the error is reduced, the variance of the measured part of the utility is increased and it is shown that, ceteris paribus, this implies a reduction in elasticity that balances the increase caused by the increase in model scale. The basic mechanism is quite simple, though the details can be complicated for particular model types. The functioning of the effect is illustrated by models estimated and applied on simulated data.
The insights offered by the paper are expected to be of help to analysts trying to understand and explain the behaviour of their models. In particular, issues have often arisen when the scale of models based on Stated Preference data are calibrated to Revealed Preference data, when it can be difficult to understand the implications of the calibration for the predicted elasticities
Association for European Transport