A Time-Series Model for Underdispersed or Overdispersed Counts

It is common for time series of unbounded counts (that is, nonnegative integers) to display overdispersion relative to the Poisson. Such an overdispersed series can be modeled by a hidden Markov model with Poisson state-dependent distributions (a “Poisson–HMM”), since a Poisson–HMM allows for both overdispersion and serial dependence. Time series of underdispersed counts seems less common, but more awkward to model; a Poisson–HMM cannot cope with underdispersion. But if in a Poisson–HMM one replaces the Poisson distributions by Conway–Maxwell–Poisson distributions, one gets a class of models which can allow for under- or overdispersion (and serial dependence). In addition, this class can cope with the combination of slight overdispersion and substantial serial dependence, a combination that is apparently difficult for a Poisson–HMM to represent. We discuss the properties of this class of models, and use direct numerical maximization of likelihood to fit a range of models to three published series of counts which display underdispersion, and to a series which displays slight overdispersion plus substantial serial dependence. In addition, we illustrate how such models can be fitted without imputation when some observations are missing from the series, and how approximate standard errors of the parameter estimates can be found. Supplementary materials for this article are available online.