GARCH Toolbox™

Likelihood Ratio Tests

Testing Support for a GARCH(2,1) Model

Example: Analysis and Estimation Using the Default Model shows that the default GARCH(1,1) model explains most of the variability of the daily returns observations of the Deutschemark/British Pound foreign-exchange rate. This example uses the lratiotest function to determine whether evidence exists to support the use of a GARCH(2,1) model. The example first fits the Deutschmark/British Pound foreign-exchange rate return series to the default GARCH(1,1) model. It then fits the same series using the following, more elaborate, GARCH(2,1) model:

1. If the Deutschmark/British Pound foreign-exchange rate data is not in your workspace, restore it:

   load garchdata
   dem2gbp = price2ret(DEM2GBP);
   

2. Estimate the GARCH(1,1) model:

   1. Create a GARCH(1,1) default model with Display set to 'off':

      spec11 = garchset('P',1,'Q',1,'Display','off');

   2. Estimate the model, including the maximized log-likelihood function value, and display the results:

      [coeff11,errors11,LLF11] = garchfit(spec11,dem2gbp);
      garchdisp(coeff11,errors11)
      Mean: ARMAX(0,0,0); Variance: GARCH(1,1)
      
      Conditional Probability Distribution: Gaussian
      Number of Model Parameters Estimated: 4
      
                                     Standard          T     
        Parameter       Value          Error       Statistic 
        ----------   -----------   ------------   -----------
                 C    -6.1919e-005   8.4331e-005     -0.7342
                 K    1.0761e-006    1.323e-007       8.1341
          GARCH(1)    0.80598        0.016561        48.6685
           ARCH(1)    0.15313        0.013974        10.9586
      

3. Estimate the GARCH(2,1) model:

   1. Create a GARCH(2,1) specification structure with Display set to 'off':

      spec21 = garchset('P',2,'Q',1,'Display','off');

   2. Then estimate the GARCH(2,1) model and display the results. Again, calculate the maximized log-likelihood function value.

      [coeff21,errors21,LLF21] = garchfit(spec21,dem2gbp);
      garchdisp(coeff21,errors21)
      Mean: ARMAX(0,0,0); Variance: GARCH(2,1)
      
      Conditional Probability Distribution: Gaussian
      Number of Model Parameters Estimated: 5
      
                                     Standard          T     
        Parameter       Value          Error       Statistic 
       -----------   -----------   ------------   -----------
                 C    -5.0071e-005   8.4756e-005     -0.5908
                 K    1.1196e-006    1.5358e-007      7.2904
          GARCH(1)    0.49404        0.11249          4.3918
          GARCH(2)    0.2938         0.10295          2.8537
           ARCH(1)    0.16805        0.016589        10.1305
      

4. Perform the Likelihood Ratio Test.

   Of the two models, GARCH(1,1) and GARCH(2,1), that are associated with the same return series:

   • The default GARCH(1,1) model is a restricted model. That is, you can interpret a GARCH(1,1) model as a GARCH(2,1) model with the restriction that G₂ = 0.

   • The more elaborate GARCH(2,1) model is an unrestricted model.

   Since garchfit enforces no boundary constraints during either of the two estimations, you can apply a likelihood ratio test (LRT) (see Hamilton [22], pages 142-144).

   In this context, the unrestricted GARCH(2,1) model serves as the alternative hypothesis; that is, the hypothesis the example gathers evidence to support. The restricted GARCH(1,1) model serves as the null hypothesis, that is, the hypothesis the example assumes is true, lacking evidence to support the alternative.

   The LRT statistic is asymptotically chi-square distributed with degrees of freedom equal to the number of restrictions imposed.

   1. Since the GARCH(1,1) model imposes one restriction, specify one degree of freedom in your call to lratiotest. Test the models at the 0.05 significance level:

      [H,pValue,Stat,CriticalValue] = lratiotest(LLF21,LLF11,...
         1,0.05);
      [H,pValue,Stat,CriticalValue]
      ans =
               1.0000    0.0218    5.2624    3.8415
      

      H = 1 indicates that there is enough statistical evidence in support of the GARCH(2,1) model.

   2. Alternatively, at the 0.02 significance level:

      [H,pValue,Stat,CriticalValue] = lratiotest(LLF21,LLF11,1,0.02);
      [H,pValue,Stat,CriticalValue]
      ans =
               0    0.0218    5.2624    5.4119
      

      H = 0 indicates that there is enough statistical evidence in support of the GARCH(2,1) model.

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Using The Autocorrelation and Partial Autocorrelation Functions

Akaike and Bayesian Information Criteria