GARCH Toolbox™

Akaike and Bayesian Information Criteria

You can use Akaike (AIC) and Bayesian (BIC) information criteria to compare alternative models. Information criteria penalize models with additional parameters. Therefore, the AIC and BIC model order selection criteria are based on parsimony (see Box, Jenkins, and Reinsel [10], pages 200-201).

The following example uses the default GARCH(1,1) and GARCH(2,1) models developed in Likelihood Ratio Tests.

1. Count the estimated parameters.

   Provide the number of parameters estimated in the model for both AIC and BIC. For the relatively simple models used here, you can just count the number of parameters. The GARCH(2,1) model estimated five parameters:

   • C

   • κ

   • G₁

   • G₂

   • A₁

   The GARCH(1,1) model estimated four parameters:

   • C

   • κ

   • G₁

   • A₁

   Tip   To count the estimated parameters in more elaborate models, use the function garchcount. garchcount accepts the output specification structure created by garchfit, and returns the number of parameters in the model defined in that structure. n21 = garchcount(coeff21) n21 = 5 n11 = garchcount(coeff11) n11 = 4

2. Compute the AIC and BIC criteria.

   1. To see the results more precisely, set the numeric format to long:

      format long
      

      Tip   You can also set the numeric format by selecting File > Preferences > Command Window > Text display from the MATLAB® toolbar.

   2. Use the aicbic function to compute the AIC and BIC statistics for the GARCH(2,1) model and the GARCH(1,1) model, and specify the number of observations in the return series:

      [AIC,BIC] = aicbic(LLF21,n21,1974);
      [AIC BIC]
      ans =
        1.0e+004 * -1.59632585502853  -1.59353194641854
      [AIC,BIC] = aicbic(LLF11,n11,1974);
      [AIC BIC]
      ans =
        1.0e+004 * -1.59599961321328  -1.59376448632528
      

      You can use the relative values of the AIC and BIC statistics as guides in the model selection process. In this example, the AIC criterion favors the GARCH(2,1) model, while the BIC criterion favors the GARCH(1,1) default model with fewer parameters. BIC imposes a greater penalty for additional parameters than does AIC. Thus, BIC always provides a given model with a number of parameters no greater than that chosen by AIC.

Likelihood Ratio Tests

Equality Constraints and Parameter Significance