"Frontmatter". In: Analysis of Financial Time Series

MISSING VALUES AND OUTLIERS 417c3t-0.4 -0.2 0.0 0.2 0.4(a) Changes in weekly interest rate1988 1990 1992 1994 1996 1998 2000year(b) Posterior probability of being an outlierprob0.0 0.2 0.4 0.6 0.81988 1990 1992 1994 1996 1998 2000year(c) Posterior mean of outlier sizemag-0.3-0.2-0.10.0 0.11988 1990 1992 1994 1996 1998 2000yearFigure 10.2. Time plots of weekly change series of U.S. 3-year Treasury constant maturityinterest rate from March 18, 1988 to September 10, 1999: (a) the data, (b) the posterior probabilityof being an outlier, and (c) the posterior mean of outlier size. The estimation is basedon a Gibbs sampling with 1050 iterations, but results of the first 50 iterations are discarded.φ 3 = 0.1, we run the Gibbs sampling for 1050 iterations, but discard results of thefirst 50 iterations. Using posterior means of the coefficients as parameter estimates,we obtain the fitted modelc 3t = 0.252c 3,t−1 + 0.003c 3,t−2 + 0.110c 3,t−2 + a t , ̂σ 2 = 0.0118,where posterior standard deviations of the parameters are 0.046, 0.045, 0.046, and0.0008, respectively. Thus, the Gibbs sampling produces results similar to that ofthe maximum likelihood method. Figure 10.2(b) shows the time plot of posteriorprobability of each observation being an additive outlier, and Figure 10.2(c) plots theposterior mean of outlier magnitude. From the probability plot, some observationshave high probabilities of being an outlier. In particular, t = 323 has a probabilityof 0.83 and the associated posterior mean of outlier magnitude is −0.304. This pointcorresponds to May 20, 1994 when the c 3t changed from 0.24 to −0.34 (i.e., abouta 0.6% drop in the weekly interest rate within two weeks). The point with second