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Example: Model Selection

As a general rule, you should specify the smallest, simplest models that adequately describe your data. This is especially relevant for estimation. Simple models are easier to estimate, easier to forecast, and easier to analyze. In fact, certain model selection criteria, such as AIC and BIC discussed in the section Model Selection, penalize models for their complexity.

Diagnostic tools such as the autocorrelation function (ACF) and partial autocorrelation function (PACF), are recommended for guiding model selection. For example, the section Example: Using the Default Model examines the ACF and PACF of the Deutschmark/British Pound foreign-exchange rate (see Time Series Data). The results support the use of a simple constant for the conditional mean model as adequate to describe the data.

The following example illustrates a complicated model specification. It simulates a return series as a pure GARCH(1,1) innovations process (the default model). It then attempts to overfit an ARMA(1,1)/GARCH(1,1) composite model to the data.

1. Create a specification structure for the innovations process and simulate the returns:

   spec = garchset('C',0,'K',0.00005,'GARCH',0.85,'ARCH',0.1,...
                   'Display','off');
   strm = RandStream('mt19937ar','Seed',5489);
   RandStream.setDefaultStream(strm);
   [e,s,y] = garchsim(spec,5000,1);

2. Fit the default model to the known GARCH(1,1) innovations process and display the estimation results:

   [coeff,errors] = garchfit(spec,y);
   garchdisp(coeff,errors)
   
   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    0.00058804     0.00040136       1.4651
              K    7.0763e-005    1.2582e-005      5.6243
       GARCH(1)    0.82343        0.021485        38.3255
        ARCH(1)    0.099154       0.011024         8.9944

   These estimation results indicate that the model that best fits the observed data is approximately

   

   

3. Continue by fitting the known GARCH(1,1) innovations process to an ARMA(1,1) mean model, and display the estimation results:

   spec11 = garchset(spec,'R',1,'M',1);
   [coeff11,errors11] = garchfit(spec11,y);
   garchdisp(coeff11,errors11)
   
      Mean: ARMAX(1,1,0); Variance: GARCH(1,1)
    
     Conditional Probability Distribution: Gaussian
     Number of Model Parameters Estimated: 6
   
                                  Standard          T     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
              C    0.00085961     0.00067665       1.2704
          AR(1)    -0.4725        0.52616         -0.8980
          MA(1)    0.49121        0.52031          0.9441
              K    7.0381e-005    1.2521e-005      5.6208
       GARCH(1)    0.82386        0.021444        38.4200
        ARCH(1)    0.099129       0.011052         8.9690

4. Examine the results. The conditional mean equation shows that the AR(1) and MA(1) parameters have similar size. In fact, when rewriting the mean equation in backshift (that is, lag) operator notation, where ,

   

   the autoregressive and moving-average polynomials come close to canceling each other (see Box, Jenkins, and Reinsel [10], pages 263-267). This is an example of parameter redundancy, or pole-zero cancellation. This supports the use of the simple default model. In fact, the more elaborate ARMA(1,1) model only complicates the analysis by requiring the estimation of two additional parameters.

Model Construction

Simulation

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