Bayesian inference and forecasting of time series under the different loss functions

We consider the different loss functions that are appropriate for the Bayesian analysis of some time series models. The Bayes inference and forecasting under these loss functions are given.;For the autoregressive model, with the Normal-Gamma and Jeffreys' priors, the posteriors are found and Bayes estimates for the parameters in the model under the different loss functions are derived, the probability density function of k-step ahead Bayes prediction is derived in a concise matrix format. In particular, Bayes estimates of the one-step ahead forecasting under the different loss functions are given. We provide the practical k-step ahead Bayesian forecasting under these loss functions. The Bayes estimates and one-step ahead and two-step ahead forecasting results under these loss functions are calculated. Under the Normal-Gamma and Jeffreys' priors, Wolfer sunspot numbers data is used to illustrate Bayes inferences and forecasts to the real life data.;For the moving average model, under the Gamma-Normal and Jeffreys' priors, based on the approximate likelihood function, the posteriors and one-step ahead forecasting probability density function are derived. Then we obtain the Bayes estimates for the parameters and predictive inferences for moving average processes under the different loss functions.