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, µ, **σ**, [φ], [θ])

**[X]**- Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
**Order**- Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Value Order 1 Ascending (the first data point corresponds to the earliest date) ( **default**).0 Descending (the first data point corresponds to the latest date). **µ**- Optional. Is the ARMA model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.
**σ**- Required. Is the standard deviation value of the model's residuals/innovations.
**[φ]**- Optional. Are the parameters of the AR(p) component model: [φ1, φ2 … φp] (starting with the lowest lag).
**[θ]**- Optional. Are the parameters of the MA(q) component model: [θ1, θ2 … θq] (starting with the lowest lag).

* *Warning

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

## Remarks

- The underlying model is described here.
- Akaike's Information Criterion (AIC) is described here.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The residuals/innovations standard deviation (σ) must 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 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.

- The number of parameters in the input argument ([φ]) determines the order of the AR component.
- The number of parameters in the input argument ([θ]) determines the order of the MA component.

## Files Examples

## Related Links

## 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: 0-471-690740.
- Box, Jenkins and Reinsel; 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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