"Frontmatter". In: Analysis of Financial Time Series

MISSING VALUES AND OUTLIERS 411y h can be classified as an additive outlier. Specifically, if y h is a value that is likelyto occur under the derived distribution, then y h is not an additive outlier. However,if the chance to observe y h is very small under the derived distribution, then y hcan be classified as an additive outlier. Therefore, detection of additive outliers andtreatment of missing values in time-series analysis are based on the same idea.In the literature, missing values in a time series can be handled by using either theKalman filter or MCMC methods; see Jones (1980) and McCulloch and Tsay (1994).Outlier detection has also been carefully investigated; see Chang, Tiao, and Chen(1988), Tsay (1988), Tsay, Peña, and Pankratz (2000), and the references therein. Theoutliers are classified into four categories depending on the nature of their impactson the time series. Here we focus on additive outliers.10.6.1 Missing ValuesFor ease in presentation, consider an AR(p) time seriesx t = φ 1 x t−1 +···+φ p x t−p + a t , (10.15)where {a t } is a Gaussian white noise series with mean zero and variance σ 2 . Supposethat the sampling period is from t = 1tot = n, but the observation x h is missing,where 1 < h < n. Our goal is to estimate the model in the presence of a missingvalue.In this particular instance, the parameters are θ = (φ ′ , x h ,σ 2 ) ′ ,whereφ =(φ 1 ,...,φ p ) ′ . Thus, we treat the missing value x h as an unknown parameter. If weassume that the prior distributions areφ ∼ N(φ o , Σ o ), x h ∼ N(µ o ,σo 2 ), vλσ 2 ∼ χ v 2 ,where the hyperparameters are known, then the conditional posterior distributionsf (φ | X, x h ,σ 2 ) and f (σ 2 | X, x h , φ) are exactly as those given in the previoussection, where X denotes the observed data. The conditional posterior distributionf (x h | X, φ,σ 2 ) is univariate normal with mean µ ∗ and variance σh 2 . These twoparameters can be obtained by using a linear regression model. Specifically, given themodel and the data, x h is only related to {x h−p ,...,x h−1 , x h+1 ,...,x h+p }. Keepingin mind that x h is an unknown parameter, we can write the relationship as follows:1. For t = h, the model saysx h = φ 1 x h−1 +···+φ p x h−p + a h .Let y h = φ 1 x h−1 +···+φ p x h−p and b h =−a h , the prior equation can bewritten aswhere φ 0 = 1.y h = x h + b h = φ 0 x h + b h ,