# GARCH_AIC - Akaike's Information Criterion (AIC) of an GARCH Model

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

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 number of parameters in the input argument - alpha - determines the order of the ARCH component model.
7. The number of parameters in the input argument - beta - determines the order of the GARCH component model.
8. GARCH(p,q) model has p+q+2 parameters to estimate.
9. 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:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
A B C D
Date Data
January 10, 2008 -2.827 GARCH(1,1)
January 11, 2008 -0.947 Mean -0.160
January 12, 2008 -0.877 Alpha_0 0.608
January 14, 2008 1.209 Alpha_1 0.00
January 13, 2008 -1.669 Beta_1 0.391
January 15, 2008 0.835
January 16, 2008 -0.266
January 17, 2008 1.361
January 18, 2008 -0.343
January 19, 2008 0.475
January 20, 2008 -1.153
January 21, 2008 1.144
January 22, 2008 -1.070
January 23, 2008 -1.491
January 24, 2008 0.686
January 25, 2008 0.975
January 26, 2008 -1.316
January 27, 2008 0.125
January 28, 2008 0.712
January 29, 2008 -1.530
January 30, 2008 0.918
January 31, 2008 0.365
February 1, 2008 -0.997
February 2, 2008 -0.360
February 3, 2008 1.347
February 4, 2008 -1.339
February 5, 2008 0.481
February 6, 2008 -1.270
February 7, 2008 1.710
February 8, 2008 -0.125
February 9, 2008 -0.940

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)