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 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. Therefore,
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 is 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 , where is the maximum likelihood estimate.
AIC = - 2( maximum log likelihood) + 2( number of free parameters)
Let be a K ×1 parameter vector that is contained in the parameter space . Given the data y, the log likelihood function is
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
Therefore,
The mean expected log likelihood function, , becomes
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, .Note that 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,
Therefore,
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.

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