"Frontmatter". In: Analysis of Financial Time Series

STOCHASTIC VOLATILITY MODELS 419that the prior distribution can be partitioned as p(β, ω) = p(β)p(ω), that is, priordistributions for the mean and volatility equations are independent. A Gibbs samplingapproach to estimating the stochastic volatility in Eqs. (10.20) and (10.21) theninvolves drawing random samples from the following conditional posterior distributions:f (β | R, X, H, ω), f (H | R, X, β, ω), f (ω | R, X, β, H).In what follows, we give details of practical implementation of the Gibbs samplingused.10.7.1 Estimation of Univariate ModelsGiven H, the mean equation in Eq. (10.20) is a nonhomogeneous linear regression.Dividing the equation by √ h t , we can write the model asr o,t = x ′ o,t β + ɛ t, t = 1,...,n, (10.22)where r o,t = r t / √ h t and x o,t = x t / √ h t , with x t = (1, x 1t ,...,x pt ) ′ being the vectorof explanatory variables. Suppose that the prior distribution of β is multivariatenormal with mean β o and covariance matrix A o . Then the posterior distribution ofβ is also multivariate normal with mean β ∗ and covariance matrix A ∗ . These twoquantities can be obtained as before via Result 1a and they are)A −1∗ =n∑t=1x o,t x ′ o,t + A−1 o , β ∗ = A ∗( n∑t=1x o,t r o,t + A −1o β owhere it is understood that the summation starts with p + 1ifr t−p is the highestlagged return used in the explanatory variables.The volatility vector H is drawn element by element. The necessary conditionalposterior distribution is f (h t | R, X, H −t , β, ω), which is produced by the normaldistribution of a t and the lognormal distribution of the volatility,f (h t | R, X, β, H −t , ω)∝ f (a t | h t , r t , x t , β) f (h t | h t−1 , ω) f (h t+1 | h t , ω)∝ ht−0.5 exp[−(r t − x ′ t β)2 /(2h t )]h −1t exp[−(ln h t − µ t ) 2 /(2σ 2 )]∝ ht −1.5 exp[−(r t − x ′ t β)2 /(2h t ) − (ln h t − µ t ) 2 /(2σ 2 )], (10.23)where µ t =[α 0 (1−α 1 )+α 1 (ln h t+1 +ln h t−1 )]/(1+α1 2) and σ 2 = σv 2/(1+α2 1 ).Herewe have used the following properties: (a) a t | h t ∼ N(0, h t ); (b) ln h t | ln h t−1 ∼N(α 0 + α 1 ln h t−1 ,σv 2);(c)lnh t+1 | ln h t ∼ N(α 0 + α 1 ln h t ,σv 2);(d)d ln h t =h −1tdh t ,whered denotes differentiation; and (e) the equality(x − a) 2 A + (x − b) 2 C = (x − c) 2 (A + C) + (a − b) 2 AC/(A + C),