Applied Bayesian Modelling - Free

218 MODELS FOR TIME SERIESThe same prior is used for the first period state choice model. For the two Poissonmeans G(1, 1) priors are stipulated, with an identifiability constraint that one is larger ±an initial run justified such a constraint, showing the means to be widely separated.With this model, a two chain run of 5000 iterations (1000 burn-in) shows a stateoccupied most of the periods (about 220 from 240), which has a low average fetalmovement rate, and a minority state with a much higher rate, around 2.2±2.3. Themajority state has a high retention rate (reflected in the transition parameter p 22 around0.96) while movement out of the minority state is much more frequent (Table 5.5).The actual number of movements is predicted closely, though Leroux and Putermanshow that using m ˆ 3 components leads to even more accurate prediction of actualcounts. The model with m ˆ 2 shows relatively small CPOs for the movements at times85 and 193 (counts of 7 and 4, respectively).For comparison, and since the outcome is a count, Model B consists of an INAR(1)model. The `innovation' process is governed by Bernoulli switching between means l 1and l 2 (with l 2 > l 1 to guarantee identifiability). Thus,Y t Poi(m t )m t ˆ p8Y t1 l 1 d t l 2 (1 d t ) t > 1m 1 ˆ l 1 d 1 l 2 (1 d 1 )with d t Bern(Z) and Z assigned a beta prior. This model also identifies a sub-populationof periods with a much higher movement rate, around 4.5, than the main set ofperiods. It has a very similar pseudo-marginal likelihood to the two-state Markovswitching model (180 vs. 179).Example 5.16 US unemployment As an illustration of models allowing both meanand variance shifts, consider the US unemployment time series analysed by RosenbergTable 5.5Lamb movements, Markov mixture model parameters and predictionsMean St. devn. 2.5% Median 97.5%p 1, 1 0.66 0.15 0.35 0.67 0.93p 1, 2 0.34 0.15 0.07 0.33 0.65p 2, 1 0.04 0.03 0.01 0.03 0.12p 2, 2 0.96 0.03 0.88 0.97 0.99Periods with s t ˆ 1 17.8 9.4 6.0 17.0 40.0Periods with s t ˆ 2 222.2 9.4 200.0 223.0 234.0l 1 2.28 0.74 1.22 2.15 4.01l 2 0.23 0.05 0.14 0.23 0.32Number of movements, actual and predicted Actual Events Predicted Events0 182 180.21 41 43.72 12 9.93 2 3.64 2 1.55 0 0.66 0 0.27 1 0.1