Probability Models for Event Counts

Abstract

The previous general introduction emphasized the need for a rich class of probability distributions when modelling count data. Since probability distributions for counts are nonstandard in the econometric literature, they are elaborated upon in this chapter. Special attention is paid to more flexible, or ‘generalized’, count data distributions since they will serve as building blocks for improved count data regression models. Furthermore, we will study properties of the underlying data generating process, where the count data may be interpreted as outcomes of an underlying count process. The classical example for a count process is the number of incoming telephone calls at a switchboard during a fixed time interval. Let the random variable N(t), t > 0, describe the number of occurences during the interval (0, t). Duration analysis studies the waiting times τ k , k = 1, 2, ..., between the (k − 1)-th and the k-th event. Count data models, by contrast, model N(T) for a given (and constant) T. By studying the relation between the underlying count process, the most prominent being the Poisson process, and the resulting probability models for event counts N, one can acquire a better understanding of the conditions under which the specific distributions are appropriate. For instance, the Poisson process, resulting in the Poisson distribution for the number of counts during a fixed time interval, requires independence and constant probabilities for the occurence of successive events, an assumption that appears overly simplistic in most applications to social sciences. Further results are derived in the course of the chapter.