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:

 
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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)

Files Examples

References

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