Referendum Models and Negative Willingness to Pay: Alternative Solutions

Traditional referendum models of contingent valuation with linear utility functions and additive errors have troubled researchers because they routinely predict negative willingness to pay. We explore solutions to this problem, including parametric models and the Turnbull empirical distribution estimator of willingness to pay exploited by Carsonet al.We show that there are closed form solutions for estimates of the empirical distribution function from the Turnbull likelihood maximization problem which correspond to the self-consistency algorithm proposed by Turnbull and the pooled adjacent violators algorithm (PAVA) described by Robertson, Wright, and Dykstra and implemented by21. Further, a lower bound estimate of willingness to pay and the associated variance can be calculated by hand. The Turnbull estimator solves the problem of estimating negative willingness to pay without resorting to ad hoc distribution assumptions. It is shown that central tendency measures of willingness to pay from parametric models are sensitive to the assumed distribution, while the lower bound Turnbull estimate is robust across distributions.