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GARCH_AIC
Calculates the Akaike's information criterion (AIC^{i}) of a given estimated GARCH^{i} model (with corrections to small sample sizes).
Syntax
X
is the univariate time series data (a one dimensional array of cells (e.g. rows or columns)).
Order
is the time order of the data series (i.e. whether the first data point corresponds to the earliest or latest date (earliest date=1 (default), latest date=0)).
Order  Description 

1  ascending (the first data point corresponds to the earliest date) 
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^{i}).
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, 2=tDistribution, 3=GED^{i}). If missing, a gaussian distribution is assumed.
value  Description 

1  (default) Gaussian or Normal Distribution 
2  Student's tDistribution 
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:
Where:
 is the loglikelihood function.
 is the number of nonmissing values.
 is the order of the ARCH component model.
 is the order of the GARCH component model.
Examples
Example 1:
A  B  C  D  

1  Date  Data  
2  January 10, 2008  2.827 
GARCH(1,1) 

3  January 11, 2008  0.947  Mean  0.160 
4  January 12, 2008  0.877  Alpha_0  0.608 
5  January 14, 2008  1.209  Alpha_1  0.00 
6  January 13, 2008  1.669  Beta_1  0.391 
7  January 15, 2008  0.835  
8  January 16, 2008  0.266  
9  January 17, 2008  1.361  
10  January 18, 2008  0.343  
11  January 19, 2008  0.475  
12  January 20, 2008  1.153  
13  January 21, 2008  1.144  
14  January 22, 2008  1.070  
15  January 23, 2008  1.491  
16  January 24, 2008  0.686  
17  January 25, 2008  0.975  
18  January 26, 2008  1.316  
19  January 27, 2008  0.125  
20  January 28, 2008  0.712  
21  January 29, 2008  1.530  
22  January 30, 2008  0.918  
23  January 31, 2008  0.365  
24  February 1, 2008  0.997  
25  February 2, 2008  0.360  
26  February 3, 2008  1.347  
27  February 4, 2008  1.339  
28  February 5, 2008  0.481  
29  February 6, 2008  1.270  
30  February 7, 2008  1.710  
31  February 8, 2008  0.125  
32  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$6,2,5)  Akaike's information criterion (AIC) for tDistribution with Freedom = 5 (103.0997)  
=GARCH_AIC($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
References
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0471690740