"Frontmatter". In: Analysis of Financial Time Series

STRUCTURE OF A MODEL 813.2 STRUCTURE OF A MODELLet r t be the log return of an asset at time index t. The basic idea behind volatilitystudy is that the series {r t } is either serially uncorrelated or with minor lower orderserial correlations, but it is dependent. For illustration, Figure 3.1 shows the ACF andPACF of some functions of the monthly log stock returns of Intel Corporation fromJanuary 1973 to December 1997. The upper left panel shows the sample ACF of thereturn, which suggests no significant serial correlations except for a minor one atlag 7. The upper right panel shows the sample ACF of the absolute log returns (i.e.,| r t |), whereas the lower left panel shows the sample ACF of the squared returnsrt 2 . These two plots clearly suggest that the monthly returns are not independent.Combining the three plots, it seems that the returns are indeed serially uncorrelated,but dependent. Volatility models attempt to capture such dependence in the returnseries.To put the volatility models in a proper perspective, it is informative to considerthe conditional mean and conditional variance of r t given F t−1 —that is,µ t = E(r t | F t−1 ), σ 2t = Var(r t | F t−1 ) = E[(r t − µ t ) 2 | F t−1 ], (3.2)Series : intelSeries : abs(intel)ACF-0.2 0.2 0.6 1.00 5 10 15 20LagACF0.0 0.4 0.80 5 10 15 20LagSeries : intel * intelSeries : intel * intelACF0.0 0.4 0.8Partial ACF-0.1 0.0 0.1 0.20 5 10 15 20Lag5 10 15 20LagFigure 3.1. Sample ACF and PACF of various functions of monthly log stock returns of IntelCorporation from January 1973 to December 1997: (a) ACF of the log returns, (b) ACF of thesquared returns (lower left), (c) ACF of the absolute returns (upper right), and (d) PACF of thesquared returns.