Probability Models for Count Data

Abstract

The introduction emphasized the need for a rich class of probability distributions when modeling count data. Since probability distributions for counts are not yet standard in the econometric literature, their properties are explored in some detail in this chapter. Special attention is paid to flexible, or ‘generalized’, count data distributions since they serve as building blocks for improved count data regression models. Furthermore, the properties of the underlying data generating process are studied. Count data frequently may be interpreted as outcomes of an underlying count process in continuous time. 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 occurrences during the interval (0,t). Duration analysis studies the waiting times τ _i , i = 1,2,..., between the (i - 1)-th and the i-th event. Count data models, by contrast, model N(T) for a given 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 a given count distribution is appropriate.