"Frontmatter". In: Analysis of Financial Time Series

24 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSsample counterpartˆρ x,y =∑ Tt=1(x t −¯x)(y t −ȳ)√ ∑Tt=1(x t −¯x) 2 ∑ Tt=1 (y t −ȳ) 2 ,where ¯x = ∑ Tt=1 x t /T and ȳ = ∑ Tt=1 y t /T are the sample mean of X and Y ,respectively.Autocorrelation Function (ACF)Consider a weakly stationary return series r t . When the linear dependence betweenr t and its past values r t−i is of interest, the concept of correlation is generalized toautocorrelation. The correlation coefficient between r t and r t−l is called the lag-lautocorrelation of r t and is commonly denoted by ρ l , which under the weak stationarityassumption is a function of l only. Specifically, we defineρ l =Cov(r t , r t−l )√Var(rt ) Var(r t−l ) = Cov(r t, r t−l )= γ l, (2.1)Var(r t ) γ 0where the property Var(r t ) = Var(r t−l ) for a weakly stationary series is used. Fromthe definition, we have ρ 0 = 1, ρ l = ρ −l ,and−1 ≤ ρ l ≤ 1. In addition, a weaklystationary series r t is not serially correlated if and only if ρ l = 0foralll>0.For a given sample of returns {r t }t=1 T ,let¯r be the sample mean (i.e., ¯r =∑ Tt=1r t /T ). Then the lag-1 sample autocorrelation of r t isˆρ 1 =∑ Tt=2(r t −¯r)(r t−1 −¯r)∑ Tt=1(r t −¯r) 2 .Under some general conditions, ˆρ 1 is a consistent estimate of ρ 1 . For example, if{r t } is an independent and identically distributed (iid) sequence and E(r 2 t )