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Figure 5-5
The rescaled residuals after
fitting the ARMA(1,1)
model of Example
5.2.5 to the lake data.
5.3 Diagnostic Checking 165
If the fitted model is appropriate, the rescaled residuals should have properties similar
to those of a WN(0, 1) sequence or of an iid(0,1) sequence if we make the stronger
assumption that the white noise {Zt} driving the ARMA process is independent white
noise.
The following diagnostic checks are all based on the expected properties of the
residuals or rescaled residuals under the assumption that the fitted model is correct
and that {Zt} ∼IID 0,σ 2 . They are the same tests introduced in Section 1.6.
5.3.1 The Graph of ˆRt,t 1,...,n
If the fitted model is appropriate, then the graph of the rescaled residuals ˆRt,t
1,...,n should resemble that of a white noise sequence with variance one. While it is
difficult to identify the correlation structure of
ˆRt (or any time series for that matter)
from its graph, deviations of the mean from zero are sometimes clearly indicated by
a trend or cyclic component and nonconstancy of the variance by fluctuations in ˆRt,
whose magnitude depends strongly on t.
The rescaled residuals obtained from the ARMA(1,1) model fitted to the meancorrected
lake data in Example 5.2.5 are displayed in Figure 5.5. The graph gives no
indication of a nonzero mean or nonconstant variance, so on this basis there is no
reason to doubt the compatibility of ˆR1,..., ˆRn with unit-variance white noise.
The next step is to check that the sample autocorrelation function of
ˆWt (or
equivalently of
ˆRt ) behaves as it should under the assumption that the fitted model
is appropriate.
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