Calculates the Akaike's information criterion (AIC) of a given estimated GARCH model (with corrections to small sample sizes).

## Syntax

**GARCH_AIC** (**[x]**, order, µ, **[α]**, [β], f, ν)

**[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 GARCH model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.
**[α]**- Required. Are the parameters of the ARCH(p) component model: [α0 α1, α2 … αp] (starting with the lowest lag).
**[β]**- Optional. Are the parameters of the GARCH(q) component model: [β1, β2 … βq] (starting with the lowest lag).
**F**- Optional. Is the probability distribution function of the innovations/residuals (1 = Gaussian (default), 2 = t-Distribution, 3 = GED).
Value Probability Distribution 1 Gaussian or Normal Distribution ( **default**).2 Student's t-Distribution. 3 Generalized Error Distribution (GED). **ν**- Optional. Is the shape parameter (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.
- For the input argument - ([α]) (parameters of the ARCH component):
- The input argument is not optional.
- The value in the first element must be positive.
- The order of the parameters starts with the lowest lag.
- One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
- In the case where alpha has one non-missing entry/element (first), no ARCH component is included.
- The order of the ARCH component model is solely determined by the order (minus one) of the last value in the array with a numeric value (vs. missing or error).
- For the input argument - ([β]) (parameters of the GARCH component):
- The input argument is optional and can be omitted, in which case no GARCH component is included.
- The order of the parameters starts with the lowest lag.
- One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
- The order of the GARCH component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
- 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.

## Files Examples

## Related Links

- Wikipedia - Akaike's Information Criterion (AIC).
- Wikipedia - Autoregressive conditional heteroskedasticity.

## References

- 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.

## Comments

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