GARCH Toolbox™

Equality Constraints and Parameter Significance

Specification Structure Fix Fields

Each of these coefficient fields in the specification structure:

• C

• AR

• MA

• Regress

• K

• GARCH

• ARCH

• Leverage

• DoF

has a corresponding logical field that lets you hold any individual parameter fixed. These fix fields are:

• FixC

• FixAR

• FixMA

• FixRegress

• FixK

• FixGARCH

• FixARCH

• FixLeverage

• FixDoF

This example fits the nasdaq return series to the default GARCH(1,1) model.

1. If the nasdaq data is not already in your workspace, restore it:

   load garchdata
   nasdaq = price2ret(NASDAQ);
   spec11 = garchset('P',1,'Q',1,'Display','off');
   [coeff11,errors11,LLF11] = garchfit(spec11,nasdaq);
   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    0.00085852     0.00018353       4.6778
              K    2.2595e-006    3.3806e-007      6.6836
       GARCH(1)    0.87513        0.0089892       97.3531
        ARCH(1)    0.11635        0.0085331       13.6348
   

2. Since the estimated model has no equality constraints, all the fixed fields are implicitly empty; for example:

   garchget(coeff11,'FixGARCH')
   ans =
        []
   

When not empty ([]), each fix field is the same size as the corresponding coefficient field.

A 0 in a particular element of a fix field indicates that the corresponding value in its companion field is an initial parameter guess. garchfit refines this guess during the estimation process.

A 1 in a particular element of a fix field indicates that garchfit holds the corresponding element of its value field fixed during the estimation process (that is, an equality constraint).

Note   To remove the constant C from the conditional mean model, that is, to fix C = 0 without providing initial parameter estimates for the remaining parameters, set C = NaN (Not-a-Number). In this case, the value of FixC has no effect.

Back to Top

Comparing the GARCH (1, 1) Estimation Results with the GARCH (2,1) Model Fit to the NASDAQ Returns

This example compares the estimation results for the default GARCH(1,1) model with those obtained from fitting a GARCH(2,1) model to the NASDAQ returns. (See Example Financial Time-Series Data Sets.)

1. Restore your workspace as needed:

   load garchdata
   nasdaq = price2ret(NASDAQ);
   

2. Estimate the model parameters and display the results:

   spec21 = garchset('P',2,'Q',1,'Display','off');
   [coeff21,errors21,LLF21] = garchfit(spec21,nasdaq);
   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    0.00086237     0.00018378       4.6925
              K    2.3016e-006    4.7519e-007      4.8436
       GARCH(1)    0.83571        0.18533          4.5092
       GARCH(2)    0.036149       0.16562          0.2183
        ARCH(1)    0.1195         0.020346         5.8734
   

   The T Statistic column is the parameter value divided by the standard error, and is normally distributed for large samples. T-statistic measures the number of standard deviations the parameter estimate is away from zero. As a general rule, a T-statistic greater than 2 in magnitude corresponds to approximately a 95 percent confidence interval. The T-statistics shown here imply that the GARCH(2) parameter adds little if any explanatory power to the model.

3. Assess significance of the GARCH(2) parameter.

   1. Constrain the GARCH(2) parameter at 0:

      specG2 = garchset(coeff21,'GARCH',[0.8 0],'FixGARCH',[0 1]);
      

      Using the specG2 structure, garchfit holds GARCH(2) fixed at 0, and refines GARCH(1) from an initial value of 0.8 during the estimation process. In other words, the specG2 specification structure tests the composite model

      

      

      

      which is mathematically equivalent to the default GARCH(1,1) model.

   2. Estimate the model subject to the equality constraint, and display the results:

      [coeffG2,errorsG2,LLFG2] = garchfit(specG2,nasdaq);
      garchdisp(coeffG2,errorsG2)
       
        Mean: ARMAX(0,0,0); Variance: GARCH(2,1)
       
        Conditional Probability Distribution: Gaussian
        Number of Model Parameters Estimated: 4
      
                                     Standard          T     
        Parameter       Value          Error       Statistic 
       -----------   -----------   ------------   -----------
                 C    0.00085827     0.00018353       4.6766
                 K    2.2574e-006    3.3785e-007      6.6818
          GARCH(1)    0.87518        0.0089856       97.3979
          GARCH(2)    0              Fixed             Fixed
           ARCH(1)    0.11631        0.0085298       13.6357
      

      The Standard Error and T-statistic columns for the second GARCH parameter indicate that garchfit holds the GARCH(2) parameter fixed. The number of estimated parameters also decreased from 5 in the original, unrestricted GARCH(2,1) model to 4 in this restricted GARCH(2,1) model. The results are virtually identical to those obtained from the GARCH(1,1) model.

   3. Apply the likelihood ratio test:

      [H,pValue,Stat,CriticalValue] = lratiotest(LLF21,LLFG2,...
                                                 1,0.05);
      [H pValue Stat CriticalValue]
      ans =
          0    0.7835   0.0755    3.8415
      

      This is the expected result. Because the two models are virtually identical, the results support acceptance of the simpler restricted model. This is essentially just the default GARCH(1,1) model.

Back to Top

Akaike and Bayesian Information Criteria

Equality Constraints and Initial Parameter Estimates