"Frontmatter". In: Analysis of Financial Time Series

408 MCMC METHODSwhich is normally distributed with mean φ and variance σ 2 ( ∑ nt=2 z 2 t−1 )−1 . Based onResult 1, the posterior distribution of φ is also normal with mean φ ∗ and varianceσ 2 ∗ ,where∑ nt=2σ∗ −2 zt−12 =σ 2 + σo −2 , φ ∗ = σ∗2(∑ nt=2z 2 t−1σ 2)̂φ + σo −2 φ o . (10.10)Finally, turn to the posterior distribution of σ 2 given β, φ, and the data. Becauseβ and φ are known, we can calculatea t = z t − φz t−1 , z t = y t − β ′ x t , t = 2,...,n.By Result 8 of Section 10.3, the posterior distribution of σ 2 is an inverted chi-squareddistribution—that is,vλ + ∑ nt=2 a 2 tσ 2 ∼ χ 2 v+(n−1) , (10.11)where χk 2 denotes a chi-squared distribution with k degrees of freedom.Using the three conditional posterior distributions in Eqs. (10.9)–(10.11), we canestimate Eq. (10.6) via Gibbs sampling as follows:1. Specify the hyperparameter values of the priors in Eq. (10.7).2. Specify arbitrary starting values for β, φ, andσ 2 (e.g., the ordinary leastsquares estimate of β without time-series errors).3. Use the multivariate normal distribution in Eq. (10.9) to draw a random realizationfor β.4. Use the univariate normal distribution in Eq. (10.10) to draw a random realizationfor φ.5. Use the chi-squared distribution in Eq. (10.11) to draw a random realizationfor σ 2 .Repeat Steps 3–5 for many iterations to obtain a Gibbs sample. The sample meansare then used as point estimates of the parameters of model (10.6).Example 10.1. As an illustration, we revisit the example of U.S. weeklyinterest rates of Chapter 2. The data are the 1-year and 3-year Treasury constantmaturity rates from January 5, 1962 to September 10, 1999 and are obtained from theFederal Reserve Bank of St Louis. Because of unit-root nonstationarity, the dependentand independent variables are1. c 3t = r 3t − r 3,t−1 , which is the weekly change in 3-year maturity rate,2. c 1t = r 1t − r 1,t−1 , which is the weekly change in 1-year maturity rate,