"Frontmatter". In: Analysis of Financial Time Series

416 MCMC METHODSFinally, the posterior distribution of β h is as follows.• If δ h = 0, then y h is not an outlier and β h ∼ N(0,ξ 2 ).• If δ h = 1, then y h is contaminated by an outlier with magnitude β h .Thevariablew j defined before contains information of β h for j = h, h + 1,...,min(h + p, n). Specifically, we have w j ∼ N(−ψ j−h β h ,σ 2 ) for j = h, h +1,...,min(h + p, n). The information can be put in a linear regression frameworkasw j =−ψ j−h β h + a j ,j = h, h + 1,...,min(h + p, n).Consequently, the information is embedded in the least squares estimate∑ mj=h−ψ j−h w ĵβ h = ∑ mj=hψ 2 , m = min(h + p, n),j−hwhich is normally distributed with mean β h and variance σ 2 /( ∑ mj=h ψ 2 j−h ).ByResult 1, the posterior distribution of β h is normal with mean βh ∗ and varianceσh∗ 2 ,whereβ ∗ h = −(∑ mj=h ψ j−h w j )ξ 2σ 2 + ( ∑ mj=h ψ 2 j−h )ξ 2 , σ2 h∗ = σ 2 ξ 2σ 2 + ( ∑ mj=h ψ 2 j−h )ξ 2 .Example 10.2. Consider the weekly change series of U.S. 3-year Treasuryconstant maturity interest rate from March 18, 1988 to September 10, 1999 for 600observations. The interest rate is in percentage and is a subseries of the dependentvariable c 3t of Example 10.1. The time series is shown in Figure 10.2(a). If ARmodels are entertained for the series, the partial autocorrelation function suggests anAR(3) model and we obtainc 3t = 0.227c 3,t−1 + 0.006c 3,t−2 + 0.114c 3,t−2 + a t , ̂σ 2 a = 0.0128,where standard errors of the coefficients are 0.041, 0.042, and 0.041, respectively.The Ljung–Box statistics of the residuals show Q(12) = 11.4, which is insignificantat the 5% level.Next we apply the Gibbs sampling to estimate the AR(3) model and to detectsimultaneously possible additive outliers. The prior distributions used areφ ∼ N(0, 0.25I 3 ),vλσ 2 = 5 × 0.00256σ 2 ∼ χ5 2 , γ 1 = 5, γ 2 = 95, ξ 2 = 0.1,where 0.00256 ≈ ̂σ 2 /5andξ 2 ≈ 9̂σ 2 . The expected number of additive outliersis 5%. Using initial values ɛ = 0.05, σ 2 = 0.012, φ 1 = 0.2, φ 2 = 0.02, and