# 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 the correction to small sample sizes).

## Syntax

ARMA_AIC(X, Order, mean, sigma, phi, theta)
X
is the univariate time series data (a 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).

Warning

ARMA_AIC() function is deprecated as of version 1.63: use ARMA_GOF function instead.

## Remarks

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

## Examples

Example 1:

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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 the ARMA model stable? (1)