ABSTRACT
In an early contribution, Harvey and Fernandes (1989) considered observations Y, which have a Poisson distribution conditional upon a mean process c,. They specify the conditional density of f: 1 conditional upon information up to time t - 1 as that of a gamma distribution with parameters a,1,_ 1 and btl1_ 1• This density corresponds to a conjugate prior for the Poisson distribution. The posterior density of E1 given observations up to time t is also gamma with parameters a1 and b,. The two sets of parameters are assumed to be linked through the relations a 111 _ 1 = wa, and b,1,_ 1 = wb1 for some 0 < w::::; I. Using this formulation the stochastic mechanism for the transition from E1_ 1 to c1 is defined implicitly. The log-likelihood for w is defined and the forecast function E(Yr+dy(T))is shown to follow an exponentially weighted moving average of past Y1 's. Similar models, based on appropriate conjugate distributions, are developed for the binomial, multinomial and negative exponential observation distributions. Harvey and Fernandes also extend their methods to incorporate estimation of explanatory variables by modifying, in the Poisson case, the mean function to E1 exp(x{ p) and apply this model to various time series of counts.
