|Time Series Analysis and Control Examples
Minimum AIC Procedure
The AIC statistic is widely used to select the
best model among alternative parametric models.
The minimum AIC model selection procedure can be interpreted
as a maximization of the expected entropy (Akaike 1981).
The entropy of a true probability density function (PDF)
with respect to the fitted PDF f is written as
where is a Kullback-Leibler
information measure, which is defined as
where the random variable Z is assumed to be continuous.
where and EZ denotes the
expectation concerning the random variable Z.
if and only if (a.s.).
The larger the quantity EZ logf(Z), the closer
the function f is to the true PDF .Given the data y = (y1, ... , yT)'
that has the same distribution as the random variable
Z, let the likelihood function of the parameter
vector be .Then the average of the log likelihood function
an estimate of the expected value of logf(Z).
Akaike (1981) derived the alternative estimate of
EZ logf(Z) by using the Bayesian predictive likelihood.
The AIC is the bias-corrected estimate of
is the maximum likelihood estimate.
be a K ×1 parameter vector that is
contained in the parameter space .
Given the data y, the log likelihood function is
AIC = - 2( maximum log likelihood) + 2( number of free parameters)
Suppose the probability density function has
the true PDF , where the true
parameter vector is contained in .
Let be a maximum likelihood estimate.
The maximum of the log likelihood function is denoted as
.The expected log likelihood function is defined by
The Taylor series expansion of the expected log
likelihood function around the true parameter
gives the following asymptotic relationship:
where is the information matrix and
= stands for asymptotic equality.
Note that since is maximized at .
By substituting , the expected
log likelihood function can be written as
The maximum likelihood estimator is asymptotically
normally distributed under the regularity conditions
The mean expected log likelihood function,
When the Taylor series expansion of the log
likelihood function around is used,
the log likelihood function is written
Since is the
maximum log likelihood function,
if the maximum likelihood estimator
is a consistent estimator of .
Replacing with the true parameter and
taking expectations with respect to the random variable Y,
Consider the following relationship:
From the previous derivation,
The natural estimator for Eis .
Using this estimator, you can write the
mean expected log likelihood function as
Consequently, the AIC is defined as an asymptotically unbiased
estimator of -2( mean expected log likelihood)
In practice, the previous asymptotic result is expected
to be valid in finite samples if the number of free
parameters does not exceed and the upper
bound of the number of free parameters is [T/2].
It is worth noting that the amount of AIC is
not meaningful in itself, since this value is
not the Kullback-Leibler information measure.
The difference of AIC values can be used to select the model.
The difference of the two AIC values is
considered insignificant if it is far less than 1.
It is possible to find a better model when the
minimum AIC model contains many free parameters.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.