Shumway Stoffer Time_Series_Analysis_and_Its_Applications__With_R_Examples 3rd edition (1)

34 1 Characteristics of Time Series
Southern Oscillation Index
ACF
−0.4 0.0 0.4 0.8
0 1 2 3 4
Recruitment
ACF
−0.2 0.2 0.6 1.0
0 1 2 3 4
SOI vs Recruitment
CCF
−0.6 −0.2 0.2
−4 −2 0 2 4
Fig. 1.14. Sample ACFs of the SOI series (top) and of the Recruitment series
(middle), and the sample CCF of the two series (bottom); negative lags indicate
SOI leads Recruitment. The lag axes are in terms of seasons (12 months).
Lag
Γ (h) = E[(x t+h − µ)(x t − µ) ′ ] (1.42)
can be defined, where the elements of the matrix Γ (h) are the cross-covariance
functions
γ ij (h) = E[(x t+h,i − µ i )(x tj − µ j )] (1.43)
for i, j = 1, . . . , p. Because γ ij (h) = γ ji (−h), it follows that
Γ (−h) = Γ ′ (h). (1.44)
Now, the sample autocovariance matrix of the vector series x t is the p × p
matrix of sample cross-covariances, defined as
n−h
∑
̂Γ (h) = n −1 (x t+h − ¯x)(x t − ¯x) ′ , (1.45)
t=1