# GARCHM_AIC - Akaike's Information Criterion (AIC) for GARCH-M

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

## Syntax

GARCHM_AIC ([x], order, µ, λ, [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.
λ
Optional. Is the volatility coefficient for the mean. In finance, lambda is referenced as the risk premium. If missing, a default of 0 is assumed.
[α]
Required. Are the parameters of the ARCH(p) component model: [αo 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

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. Given a fixed data set, several competing models may be ranked according to their AIC, the model with the lowest AIC being the best.
6. The GARCH-M (p, q) model with Gaussian has p+q+3 estimated parameters.
7. The number of parameters in the input argument - [αo α1, α2 … αp] - determines the order of the ARCH component model.
8. The number of parameters in the input argument - [β1, β2 … βq] - determines the order of the GARCH component model.