"Frontmatter". In: Analysis of Financial Time Series

410 MCMC METHODS0 5 10 15-0.05 0.0 0.05 0.10b00 1 2 3 4 50.1 0.2 0.3 0.4 0.5 0.6phi10 20 40 60 80 100 120 1400.070 0.075 0.080sigma0 10 20 30 400.76 0.78 0.80 0.82b10 1 2 3 4 5-0.1 0.0 0.1 0.2 0.3phi2Figure 10.1. Histograms of Gibbs draws for the model in Eq. (10.13) with 2100 iterations.The results are based on the last 2000 draws. Prior distributions and starting parameter valuesare given in the text.10.6 MISSING VALUES AND OUTLIERSIn this section, we discuss MCMC methods for handling missing values and detectingadditive outliers. Let {y t } n t=1 be an observed time series. A data point y h is anadditive outlier if{xh + ω if t = hy t =(10.14)otherwise,x twhere ω is the magnitude of the outlier and x t is an outlier-free time series. Examplesof additive outliers include recording errors (e.g., typos and measurement errors).Outliers can seriously affect time-series analysis because they may induce substantialbiases in parameter estimation and lead to model misspecification.Consider a time series x t and a fixed time index h. We can learn a lot about x h bytreating it as a missing value. If the model of x t were known, then we could derivethe conditional distribution of x h given the other values of the series. By comparingthe observed value y h with the derived distribution of x h , we can determine whether