"Frontmatter". In: Analysis of Financial Time Series

STOCHASTIC VOLATILITY MODELS 421where η = α 0 /(1 − α 1 ) and | α 1 | < 1. The model of the reversed series is(ln h t − η) = α 1 (ln h t+1 − η) + v ∗ t ,where {v ∗ t } is also a Gaussian white noise series with mean zero and variance σ 2 v .Consequently, the 2-step backward prediction of h 0 at time t = 2isĥ 2 (−2) = α 2 1 (ln h 2 − η).Remark: The formula (10.23) can also be obtained by using results of a missingvalue in an AR(1) model; see subsection 10.6.1. Specifically, assume that ln h t ismissing. For the AR(1) model in Eq. (10.21), this missing value is related to ln h t−1and ln h t+1 for 1 < t < n. From the model, we haveln h t = α 0 + α 1 ln h t−1 + a t .Define y t = α 0 + α 1 ln h t−1 , x t = 1, and b t =−a t . Then we obtainNext, fromy t = x t ln h t + b t . (10.24)ln h t+1 = α 0 + α 1 ln h t + a t+1 ,we define y t+1 = ln h t+1 − α 0 , x t+1 = α 1 and b t+1 = a t+1 , and obtainy t+1 = x t+1 ln h t + b t+1 . (10.25)Now Eqs. (10.24) and (10.25) form a special simple linear regression with two observationsand an unknown parameter ln h t . Note that b t and b t+1 have the same distributionbecause −a t is also N(0,σ 2 v ). The least squares estimate of ln h t is thenln ̂h t = x t y t + x t+1 y t+1xt 2 + xt+12= α 0(1 − α 1 ) + α 1 (ln h t+1 + ln h t−1 )1 + α12 ,which is precisely the conditional mean of ln h t given in Eq. (10.23). In addition, thisestimate is normally distributed with mean ln h t and variance σv 2/(1 + α2 1). Formula(10.23) is simply the product of a t ∼ N(0, h t ) and ln ̂h t ∼ N[ln h t ,σv 2/(1 + α2 1 )]with the transformation d ln h t = h −1t dh t . This regression approach generalizes easilyto other AR(p) models for ln h t . We use this approach and assume that {h t } p t=1are fixed for a stochastic volatility AR(p) model.Remark: Starting values of h t can be obtained by fitting a volatility model ofChapter 3 to the return series.