# ARMA_AIC - Akaike's Information Criterion (AIC) of an ARMA Model

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

1. The underlying model is described here.
2. Akaike's Information Criterion (AIC) is described here.
3. Warning: ARMA_AIC() function is deprecated as of version 1.63: use ARMA_GOF function instead.
4. The time series is homogeneous or equally spaced.
5. The time series may include missing values (e.g. #N/A) at either end.
6. The standard deviation (i.e. $\sigma$) of the ARMA model's residuals should be greater than zero.
7. Given a fixed data set, several competing models may be ranked according to their AIC, the model with the lowest AIC being the best.
8. The ARMA model has p+q+2 parameters, and it has independent and normally distributed residuals with constant variance.
9. Maximizing for the log-likelihood 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 non-missing 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.
10. The number of parameters in the input argument - phi - determines the order of the AR component.
11. The number of parameters in the input argument - theta - determines the order of the MA component.

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B C D
Date Data
1/1/2008 -0.300 ARMA
1/2/2008 -1.278 Mean -0.00258
1/3/2008 0.244 Sigma 0.14
1/4/2008 1.276 Phi_1 -0.236
1/6/2008 1.733 Theta_1 -5.60E-05
1/7/2008 -2.184
1/8/2008 -0.234
1/9/2008 1.095
1/10/2008 -1.087
1/11/2008 -0.690
1/12/2008 -1.690
1/13/2008 -1.847
1/14/2008 -0.978
1/15/2008 -0.774

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$6) Log-Likelihood Function (-519.095)
=ARMA_CHECK($D$3,$D$4,$D$5,$D$6) Is ARMA model stable? (1)