GARCH Toolbox

Ordinary Least Squares Regression

The following example illustrates an ordinary least squares regression by simulating a return series that scales the daily return values of the New York Stock Exchange Composite Index. It also provides an example of a constant conditional variance model.

1. Load the NYSE data set and convert the price series to a return series.

• load garchdata
  nyse = price2ret(NYSE);
  

2. Create a specification structure. Set the Display flag to 'off' to suppress the optimization details that garchfit normally displays.

• spec = garchset('P',0,'Q',0,'C',0,'Regress',1.2,'K',0.00015,...
                  'Display', 'off')
  
  spec = 
            Comment: 'Mean: ARMAX(0,0,?); Variance: GARCH(0,0)'
       Distribution: 'Gaussian'
                  C: 0
            Regress: 1.2000
      VarianceModel: 'GARCH'
                  K: 1.5000e-004
            Display: 'off'
  

3. Simulate a single realization of 2000 observations, fit the model, and examine the results.

• [e,s,y] = garchsim(spec,2000,1,[],nyse);
  [coeff,errors] = garchfit(spec,y,nyse);
  garchdisp(coeff,errors)
   
    Mean: ARMAX(0,0,1); Variance: GARCH(0,0)
   
    Conditional Probability Distribution: Gaussian
    Number of Model Parameters Estimated: 3
  
                                 Standard          T     
    Parameter       Value          Error       Statistic 
   -----------   -----------   ------------   -----------
             C    4.9091e-006    0.00027114       0.0181
    Regress(1)    1.2251         0.028909        42.3786
             K    0.00014662     4.6945e-006     31.2334
  

1. These estimation results are just the ordinary least squares (OLS) regression results. In fact, in the absence of GARCH effects and assuming Gaussian innovations, maximum likelihood estimation and least squares regression are the same thing.

Note    This example is shown purely for illustration purposes. Although you can use the GARCH Toolbox to perform OLS, it is computationally inefficient and is not recommended.

Generating Forecasted Explanatory Data

Regression in a Monte Carlo Framework