A Theoretical Foundation for Count Data Models

For several decades, economists have used the annual demand for trips in order to measure the nonmarket value of recreation sites. Two features of trip demand functions complicate their estimation: trip demand is nonnegative and occurs in integer quantities. The fact that trip demand cannot be negative results in a censored (at zero) data set; failure to account for censoring leads to biased estimation. The integer nature of trip demand, when continuous models are estimated, can also lead to biased results. A variety of techniques have been developed to deal with these problems, including models incorporating truncated error distributions, random utility models, discrete/continuous models, and repeated discrete choice models.' In this paper we explore the use of count data estimators, such as the Poisson model, to embody the recreational demand for trips. Poisson models are becoming increasingly common (Hellerstein, Creel, and Loomis; Smith, Shaw, Terza, and Wilson). In addition, a variety of count model extensions to the Poisson have been recently developed, providing analysts with a menu of robust and flexible estimators (see Hausman, Hall, and Griliches, or Cameron and Trivedi). Although the attractive econometric properties of count estimators are well understood, a theoretical foundation for their use in welfare analysis has not yet been presented. In particular, the link between an individual consumer's optimization problem and a count estimator has not been drawn. Without a theoretical foundation, it is not clear how to interpret count models. More importantly, it remains ambiguous how to apply the results of count estimators of demand to welfare analysis. For example, it is unclear how to value recreation sites on the basis of a count demand model for trips. Our paper addresses this shortcoming by developing two theoretical frameworks for count demand models. The first model modifies a standard, continuous demand model to account for a constrained integer choice set. The second approach is based on a discrete choice model which is then repeated over time. Welfare measures based on both these underlying models are derived, and are shown to yield the same formula for measuring consumer surplus.