Assuring Finite Moments for Willingness To Pay in Random Coefficient Models

01 April 2011
By Dr. Kenneth Train et al.

Random coefficient models such as mixed logit are increasingly being used to allow for random heterogeneity in willingness to pay (WTP) measures. In the most commonly used specifications, the distribution of WTP for an attribute is derived from the distribution of the ratio of individual coefficients. Since the cost coefficient enters the denominator, its distribution plays a major role in the distribution of WTP. Depending on the choice of distribution for the cost coefficient, and its implied range, the distribution of WTP may or may not have finite moments. In this paper, the authors identify a criterion to determine whether, with a given distribution for the cost coefficient, the distribution of WTP has finite moments. Using this criterion, the authors show that some popular distributions used for the cost coefficient in random coefficient models, including normal, truncated normal, uniform and triangular, imply infinite moments for the distribution of WTP, even if truncated or bounded at zero. The authors also point out that relying on simulation approaches to obtain moments of WTP from the estimated distribution of the cost and attribute coefficients can mask the issue by giving finite moments when the true ones are infinite.

This paper was published in Transportation, Vol. 39, No. 1, pp. 19-31, 2011.