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, 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).

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 residuals/innovations standard deviation (σ) must 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 ([φ]) determines the order of the AR component.
10. The number of parameters in the input argument ([θ]) determines the order of the MA component.

Files Examples

Related Links

• Wikipedia - Akaike's Information Criterion (AIC).
• Wikipedia - Autoregressive moving average model.

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.