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5.5 Order Selection 171
Table 5.2 ˆσ 2 p and FPEp for AR(p) models fitted
to the lake data.
p σ 2 p FPEp
0 1.7203 1.7203
1 0.5097 0.5202
2 0.4790 0.4989
3 0.4728 0.5027
4 0.4708 0.5109
5 0.4705 0.5211
6 0.4705 0.5318
7 0.4679 0.5399
8 0.4664 0.5493
9 0.4664 0.5607
10 0.4453 0.5465
To apply the FPE criterion for autoregressive order selection we therefore choose
the value of p that minimizes FPEp as defined in (5.5.2).
Example 5.5.1 FPE-based selection of an AR model for the lake data
In Example 5.1.4 we fitted AR(2) models to the mean-corrected lake data, the order 2
being suggested by the sample PACF shown in Figure 5.4. To use the FPE criterion to
select p, we have shown in Table 5.2 the values of FPE for values of p from 0 to 10.
These values were found using ITSM by fitting maximum likelihood AR models with
the option Model>Estimation>Max likelihood. Also shown in the table are the
values of the maximum likelihood estimates of σ 2 for the same values of p. Whereas
ˆσ 2 p decreases steadily with p, the values of FPEp have a clear minimum at p 2,
confirming our earlier choice of p 2 as the most appropriate for this data set.
5.5.2 The AICC Criterion
A more generally applicable criterion for model selection than the FPE is the information
criterion of Akaike (1973), known as the AIC. This was designed to be an
approximately unbiased estimate of the Kullback–Leibler index of the fitted model
relative to the true model (defined below). Here we use a bias-corrected version of
the AIC, referred to as the AICC, suggested by Hurvich and Tsai (1989).
If X is an n-dimensional random vector whose probability density belongs to
the family {f(·; ψ),ψ ∈ }, the Kullback–Leibler discrepancy between f(·; ψ) and
f(·; θ) is defined as
d(ψ|θ) (ψ|θ)− (θ|θ),