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
- 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 ARMA model's residuals' standard deviation (i.e., $\sigma$) 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 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 - 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:
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|
| 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) |
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 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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