"Frontmatter". In: Analysis of Financial Time Series

414 MCMC METHODSUnder the model in Eq. (10.18), we have n data points, but there are 2n + p + 3parameters—namely, φ = (φ 0 ,...,φ p ) ′ , δ = (δ 1 ,...,δ n ) ′ , β = (β 1 ,...,β n ) ′ , σ 2 ,and ɛ. The binary parameters δ t are governed by ɛ and β t s are determined by thespecified distribution. The parameters δ and β are introduced by using the idea ofdata augmentation with δ t denoting the presence or absence of an additive outlier attime t, andβ t is the magnitude of the outlier at time t when it is present.Assume that the prior distributions arevλφ ∼ N(φ o , Σ o ),σ 2 ∼ χ v 2 , ɛ ∼ beta(γ 1,γ 2 ), β t ∼ N(0,ξ 2 ),where the hyperparameters are known. These are conjugate prior distributions. Toimplement Gibbs sampling for model estimation with outlier detection, we need toconsider the conditional posterior distributions off (φ | Y, δ, β,σ 2 ), f (δ h | Y, δ −h , β, φ,σ 2 ), f (β h | Y, δ, β −h , φ,σ 2 ),f (ɛ | Y, δ), f (σ 2 | Y, φ, δ, β),where 1 ≤ h ≤ n, Y denotes the data, and θ −i denotes that the ith element of θ isremoved.Conditioned on δ and β, the outlier-free time series x t can be obtained by x t =y t − δ t β t . Information of the data about φ is then contained in the least squaresestimate( ) n∑−1 ( )n∑̂φ = x t−1 x ′ t−1x t−1 x t ,t=p+1t=p+1where x t−1 = (1, x t−1 ,...,x t−p ) ′ , which is normally distributed with mean φ andcovariance matrix( n∑−1̂Σ = σ 2 x t−1 x t−1) ′ .t=p+1The conditional posterior distribution of φ is therefore multivariate normal withmean φ ∗ and covariance matrix Σ ∗ , which are given in Eq. (10.9) with β beingreplaced by φ and x o,t by x t−1 . Similarly, the conditional posterior distribution ofσ 2 is an inverted chi-squared distribution—that is,vλ + ∑ nt=p+1 at2σ 2 ∼ χv+(n−p) 2 ,where a t = x t − φ ′ x t−1 and x t = y t − δ t β t .The conditional posterior distribution of δ h can be obtained as follows. First, δ h isonly related to {y j ,β j } h+pj=h−p , {δ j} h+pj=h−p with j ̸= h, φ, andσ 2 . More specifically,