In Example 6.6 (benefit claimants), re-estimate the INAR models to include the a latent preseries data point, by setting a prior \(y_{0} \sim \operatorname{Po}\left(\omega_{0}\right)\), where \(\omega_{0}\) may be gamma, \(\log\)-normal, etc.

Data from Example 6.6

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Consider monthly data on injury claimants for cuts and lacerations analysed by Freeland and McCabe (2004), Enciso-Mora et al. (2009) and others. There are T = 120 observations between January 1985 and December 1994. Here an INAR lag-1 model with and without sea- sonal covariates is estimated first. Without covariates one has a constant mean , and data generating process y=py-1+w, w ~ Po(2), here with priors ~Ga(1,0.001), and p ~ Be(1, 1). Denoting S = py-1 the likelihood is T 1=2 The analysis conditions on the first observation, though it would be possible to set an initialising Poisson prior on yo In BUGS, the code for this model constrains the binomial survival component at time t to be less than the value of y. The remaining part of the likelihood, the Poisson part for the innovations w = y-pyt-1=y- S, is specified using the new sampling distribution option, which in WinBUGS results in