"Frontmatter". In: Analysis of Financial Time Series

418 MCMC METHODShighest posterior probability of being an outlier is t = 201, which is January 17,1992. The outlying posterior probability is 0.58 and the estimated outlier size is0.176. At this second point, c 3t changed from −0.02 to 0.33, corresponding to ajump of about 0.35% in the weekly interest rate.Remark: Outlier detection via Gibbs sampling requires intensive computation,but the approach performs a joint estimation of model parameters and outliers. Yetthe traditional approach to outlier detection separates estimation from detection. Itis much faster in computation, but may produce spurious detections when multipleoutliers are present. For the data in Example 10.2, the SCA program also identifiest = 323 and t = 201 as the two most significant additive outliers. The estimatedoutlier sizes are −0.39 and 0.36, respectively.10.7 STOCHASTIC VOLATILITY MODELSAn important financial application of MCMC methods is the estimation of stochasticvolatility models; see Jacquier, Polson, and Rossi (1994) and the references therein.We start with a univariate stochastic volatility model. The mean and volatility equationsof an asset return r t arer t = β 0 + β 1 x 1t +···+β p x pt + a t , a t = √ h t ɛ t (10.20)ln h t = α 0 + α 1 ln h t−1 + v t (10.21)where {x it | i = 1,...,p} are explanatory variables available at time t − 1, β j sare parameters, {ɛ t } is a Gaussian white noise sequence with mean 0 and variance 1,{v t } is also a Gaussian white noise sequence with mean 0 and variance σv 2,and{ɛ t}and {v t } are independent. The log transformation is used to ensure that h t is positivefor all t. The explanatory variables x it may include lagged values of the return (e.g.,x it = r t−i ). In Eq. (10.21), we assume that | α 1 | < 1 so that the log volatility processln h t is stationary. If necessary, a higher order AR(p) model can be used for ln h t .Denote the coefficient vector of the mean equation by β = (β 0 ,β 1 ,...,β p ) ′ andthe parameter vector of the volatility equation by ω = (α 0 ,α 1 ,σv 2)′ . Suppose thatR = (r 1 ,...,r n ) ′ is the collection of observed returns and X is the collection ofexplanatory variables. Let H = (h 1 ,...,h n ) ′ be the vector of unobservable volatilities.Here β and ω are the “traditional” parameters of the model and H is an auxiliaryvariable. Estimation of the model would be complicated via the maximum likelihoodmethod because the likelihood function is a mixture over the n-dimensional H distributionas∫f (R | X, β, ω) = f (R | X, β, H) f (H | ω)dH.However, under the Bayesian framework, the volatility vector H consists of augmentedparameters. Conditioning on H, we can focus on the probability distributionfunctions f (R | H, β) and f (H | ω) and the prior distribution p(β, ω). We assume