"Frontmatter". In: Analysis of Financial Time Series

FORECASTING 439garchrtn-sq0 200 400 6000 500 1000 1500 2000(a) squared log returns1940 1960year1980 2000(b) a garch-m model1940 1960year1980 2000(c) A Markov switching garch-m modelmsw0 200 400 6001940 1960 1980 2000yearFigure 10.15. Fitted volatility series for the monthly log returns of GE stock from 1926 to1999: (a) The squared log returns, (b) the GARCH-M model in Eq. (10.42), and (c) the twostateMarkov switching GARCH-M model in Eq. (10.40).Consider the stochastic volatility model in Eqs. (10.20) and (10.21). Suppose thatthere are n returns available and we are interested in predicting the return r n+i andvolatility h n+i for i = 1,...,l,wherel>0. Assume that the explanatory variablesx jt in Eq. (10.20) are either available or can be predicted sequentially during theforecasting period. Recall that estimation of the model under the MCMC frameworkis done by Gibbs sampling, which draws parameter values from their conditionalposterior distributions iteratively. Denote the parameters by β j = (β 0, j ,...,β p, j ) ′ ,α j = (α 0, j ,α 1, j ) ′ ,andσ 2 v, jfor the jth Gibbs iteration. In other words, at the jthGibbs iteration, the model isr t = β 0, j + β 1, j x 1t +···+β p, j x pt + a t (10.43)ln h t = α 0, j + α 1, j ln h t−1 + v t , Var(v t ) = σ 2 v, j . (10.44)We can use this model to generate a realization of r n+i and h n+i for i = 1,...,l.Denote the simulated realizations by r n+i, j and h n+i, j , respectively. These realizationsare generated as follows: