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## A First Course on Time Series Analysis : Examples with SAS by Michael Falk PDF

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Is called autocovariance function and ρ(k) := γ(k) , γ(0) k = 0, 1, . . is called autocorrelation function. Let y1 , . . , yn be realizations of a time series Y1 , . . , Yn . The empirical counterpart of the autocovariance function is 1 c(k) := n n−k n 1 (yt+k − y¯)(yt − y¯) with y¯ = n t=1 yt t=1 and the empirical autocorrelation is defined by c(k) r(k) := = c(0) n−k ¯)(yt − t=1 (yt+k − y n ¯)2 t=1 (yt − y y¯) . 8 (ii) for the particular role of the factor 1/n in place of 1/(n − k) in the definition of c(k).

3 Autocovariances and Autocorrelations Autocovariances and autocorrelations are measures of dependence between variables in a time series. Suppose that Y1 , . . , Yn are square integrable random variables with the property that the covariance Cov(Yt+k , Yt ) = E((Yt+k − E(Yt+k ))(Yt − E(Yt ))) of observations with lag k does not depend on t. Then γ(k) := Cov(Yk+1 , Y1 ) = Cov(Yk+2 , Y2 ) = . . is called autocovariance function and ρ(k) := γ(k) , γ(0) k = 0, 1, . . is called autocorrelation function.

T) = 0 for t = 0. 7. Suppose that Yt = u au εt−u , t ∈ Z, is a general linear process with u |au ||z u | < ∞, if r −1 < |z| < r for some r > 1. Put σ 2 := Var(ε0 ). The process (Yt ) then has the covariance generating function G(z) = σ 2 au z u a ¯u z −u , u u r−1 < |z| < r. Proof. 6 implies for t ∈ Z Cov(Yt , Y0 ) = u 2 w =σ au a ¯w γε (t + w − u) au a ¯u−t . 1 Linear Filters and Stochastic Processes 53 This implies G(z) = σ 2 au a ¯u−t z t t u = σ2 u = σ2 u |au |2 + |au |2 + au a ¯u−t z t + t≥1 u t≤−1 u au a ¯t z u−t + u t≤u−1 u a ¯t z −t .