Calculates the Akaike's information criterion (AIC) of the given estimated ARMA model (with correction to small sample sizes).
Syntax
ARMA_AIC(X, Order, mean, sigma, phi, theta)
X is the univariate time series data (one dimensional array of cells (e.g. rows or columns)).
Order is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).
Order  Description 

1  ascending (the first data point corresponds to the earliest date) (default) 
0  descending (the first data point corresponds to the latest date) 
mean is the ARMA model mean (i.e. mu).
sigma is the standard deviation of the model's residuals/innovations.
phi are the parameters of the AR(p) component model (starting with the lowest lag).
theta are the parameters of the MA(q) component model (starting with the lowest lag).
Remarks
 The underlying model is described here.
 Akaike's Information Criterion (AIC) is described here.
 Warning: ARMA_AIC() function is deprecated as of version 1.63: use ARMA_GOF function instead.
 The time series is homogeneous or equally spaced.
 The time series may include missing values (e.g. #N/A) at either end.
 The standard deviation (i.e. $\sigma$) of the ARMA model's residuals should be greater than zero.
 Given a fixed data set, several competing models may be ranked according to their AIC, the model with the lowest AIC being the best.
 The ARMA model has p+q+2 parameters, and it has independent and normally distributed residuals with constant variance.
 Maximizing for the loglikelihood function, the AICc function for an ARMA model becomes:
$$\mathit{AICc}(p,q)= \ln(\hat\sigma^2(p,q))+\frac{2\times(p+q)}{T}$$
Where:
 $T$ is the number of nonmissing values in the time series.
 $p$ is the order of the AR component model.
 $q$ os the order of the MA component model.
 $\hat\sigma$ is the standard deviation of the residuals.
 The number of parameters in the input argument  phi  determines the order of the AR component.
 The number of parameters in the input argument  theta  determines the order of the MA component.
Examples
Example 1:


Formula  Description (Result)  

=ARMA_AIC($B$2:$B$15,1,$D$3,$D$4,$D$5,$D$6)  Akaike's Information Criterion (1046.59)  
=ARMA_LLF($B$2:$B$15,1,$D$3,$D$4,$D$5,$D$6)  LogLikelihood Function (519.095)  
=ARMA_CHECK($D$3,$D$4,$D$5,$D$6)  Is ARMA model stable? (1) 
Files Examples
References
 D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
 James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
 Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0471690740
 Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848
 Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568
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