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62 Chapter 2 Stationary Processes
ACF
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Figure 2-1
The sample autocorrelation
function of n 200
observations of the MA(1)
process in Example 2.4.3,
showing the bounds
±1.96n−1/2 (1 + 2ˆρ 2 (1)) 1/2 0 10 20 30 40
. Lag
−.4333 −6.128n −1/2 , which would cause us (in the absence of our prior knowledge
of {Xt}) to reject the hypothesis that the data are a sample from an iid noise sequence.
The fact that |ˆρ(h)| ≤1.96n −1/2 for h 2,...,40 strongly suggests that the data are
from a model in which observations are uncorrelated past lag 1. In Figure 2.1 we have
plotted the bounds ±1.96n −1/2 (1+2ρ 2 (1)) 1/2 , indicating the compatibility of the data
with the model (2.4.11). Since, however, ρ(1) is not normally known in advance, the
autocorrelations ˆρ(2),..., ˆρ(40) would in practice have been compared with the more
stringent bounds ±1.96n −1/2 or with the bounds ±1.96n −1/2 (1+2 ˆρ 2 (1)) 1/2 in order to
check the hypothesis that the data are generated by a moving-average process of order
1. Finally, it is worth noting that the lag-one correlation −.4878 is well inside the 95%
confidence bounds for ρ(1) given by ˆρ(1) ± 1.96n −1/2 (1 − 3 ˆρ 2 (1) + 4 ˆρ 4 (1)) 1/2
−.4333 ± .1053. This further supports the compatibility of the data with the model
Xt Zt − 0.8Zt−1.
Example 2.4.4 An AR(1) process
For the AR(1) process of Example 2.2.1,
Xt φXt−1 + Zt,
where {Zt} is iid noise and |φ| < 1, we have, from (2.4.10) with ρ(h) φ |h| ,
wii
i
φ 2i φ −k − φ k2 +
k1
∞
ki+1
φ 2k φ −i − φ i 2
1 − φ 2i 1 + φ 2 1 − φ 2 −1 − 2iφ 2i , (2.4.12)