Calculates the Akaike's information criterion (AIC) of a given estimated GARCH model (with corrections to small sample sizes).
Syntax
GARCH_AIC(X, Order, mean, alphas, betas, innovation, v)
- 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 GARCH model mean (i.e. mu).
- alphas
- are the parameters of the ARCH(p) component model (starting with the lowest lag).
- betas
- are the parameters of the GARCH(q) component model (starting with the lowest lag).
- innovation
- is the probability distribution function of the innovations/residuals (1=Gaussian (default), 2=t-Distribution, 3=GED).
value Description 1 (default) Gaussian or Normal Distribution 2 Student's t-Distribution 3 Generalized Error Distribution (GED) - v
- is the shape factor (or degrees of freedom) of the innovations/residuals probability distribution function.
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.
- 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 number of parameters in the input argument - alpha - determines the order of the ARCH component model.
- The number of parameters in the input argument - beta - determines the order of the GARCH component model.
- GARCH(p,q) model has p+q+2 parameters to estimate.
- The AIC for a GARCH model is defined as:
$AIC = 2(p+q+1) - 2\times \ln L^*$
Where:
- $\ln L^*$ is the log-likelihood function.
- $T$ is the number of non-missing values.
- $p$ is the order of the ARCH component model.
- $q$ is the order of the GARCH component model.
Examples
Example 1:
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| Formula | Description (Result) |
|---|---|
| =GARCH_AIC($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6) | Akaike's information criterion (AIC) for Guassian Distribution (96.013) |
| =GARCH_AIC($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6,2,5) | Akaike's information criterion (AIC) for t-Distribution with Freedom = 5 (103.0997) |
| =GARCH_AIC($B$2:$B$32,1+$D$14,$D$3,$D$4:$D$5,$D$6,3,2) | Akaike's information criterion (AIC) for GED with Freedom = 2 (96.013) |
Files Examples
Related Links
References
- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
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