Forecasting under the MCMC framework can be done easily. The procedure is simply to use the fitted model in each Gibbs iteration to generate samples for the forecasting period. In a sense, forecasting here is done by using the fitted model to simulate realizations for the forecasting period. We use the univariate stochastic volatility model to illustrate the procedure; forecasts of other models can be obtained by the same method.
Consider the stochastic volatility model in Eqs. (12.20) and (12.21). Suppose that there are n returns available and we are interested in predicting the return rn+i and volatility hn+i for i = 1, … , ℓ, where ℓ > 0. Assume that the explanatory variables xjt in Eq. (12.20) are either available or can be predicted sequentially during the forecasting period. Recall that estimation of the model under the MCMC framework is done by Gibbs sampling, which draws parameter values from their conditional posterior distributions iteratively. Denote the parameters by , , and for the jth Gibbs iteration. In other words, at the jth Gibbs iteration, the model is
We can use this model to generate a realization of rn+i and hn+i for i = 1, … , ℓ. ...