GARCH Toolbox

Simulating Sample Paths

The section Analysis and Estimation Example Using the Default Model models the equity series of a hypothetical company, the XYZ Corporation, using the default model. This section uses the resulting model

to simulate sample paths, using the simulation function garchsim, for return series, innovations, and conditional standard deviation processes. You can think of garchsim as a filter that you can use to generate a (possibly) correlated return series {y_t} from a white noise input series {_t}.

Use the following commands to restore your workspace if necessary. This example omits the estimation output to save space.

• load xyz
  xyz = price2ret(prices);
  [coeff, errors, LLF, innovations, sigma] = garchfit(xyz);
  coeff
  coeff = 
           Comment: 'Mean: ARMAX(0,0,0); Variance: GARCH(1,1)'
                 R: 0
                 M: 0
                 P: 1
                 Q: 1
      Distribution: 'Gaussian'
                 C: 4.9183e-004
                AR: []
                MA: []
           Regress: []
                 K: 8.2736e-007
             GARCH: 0.9628
              ARCH: 0.0318
              FixC: []
             FixAR: []
             FixMA: []
        FixRegress: []
              FixK: []
          FixGARCH: []
           FixARCH: []
      Optimization: [1x1 struct]
  

Using Default Inputs

Now call garchsim to simulate sample paths using the model in coeff. This command accepts garchsim defaults for:

• The number of sample paths (i.e. realizations) to generate: 1
• The number of observations to generate for each path: 100
• The random number generator seed: 0

• [e,s,y] =  garchsim(coeff);
  
  whos e s y
    Name      Size         Bytes  Class
  
    e       100x1            800  double array
    s       100x1            800  double array
    y       100x1            800  double array
  
  Grand total is 300 elements using 2400 bytes
  

The result is a single realization (i.e., one sample path) of 100 observations each for the innovations {_t}, conditional standard deviations {_t}, and returns {y_t} processes. These processes are designated by the output variables e, s, and y, respectively.

Simulating a Much Longer Path

However, accurate GARCH modeling typically requires a few years worth of data. If there are 250 trading days per year, 1000 observations would be a more useful sample.

• [e,s,y] =  garchsim(coeff, 1000);
  whos e s y
    Name      Size         Bytes  Class
  
    e      1000x1           8000  double array
    s      1000x1           8000  double array
    y      1000x1           8000  double array
  
  Grand total is 3000 elements using 24000 bytes
  

The result is a single realization of 1000 observations (roughly four years of data) for each of {_t}, {_t}, and {y_t}. Plot the garchsim output data to see what it looks like.

• garchplot(e,s,y)
  
  
  

Figure 2-14: A Single Realization of 1000 Observations

Simulating Multiple Paths

However, Monte Carlo simulation requires multiple independent paths. Use the same model to simulate 1000 paths of 200 observations each.

• [e,s,y] =  garchsim(coeff, 200, 1000);
  whos e s y
    Name      Size         Bytes  Class
  
    e       200x1000     1600000  double array
    s       200x1000     1600000  double array
    y       200x1000     1600000  double array
  
  Grand total is 600000 elements using 4800000 bytes
  

In this example, {_t}, {_t}, and {y_t} are 200-by-1000 element matrices. These are relatively large arrays, and demand large chunks of memory. In fact, because of the way the GARCH Toolbox manages transients, simulating this data requires more memory than the 4800000 bytes indicated above.

Simulation

Transients in the Simulation Process