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The AUTOREG Procedure

Example 8.6: Estimation of ARCH(2) Process

Stock returns show a tendency for small changes to be followed by small changes while large changes are followed by large changes. The plot of daily price changes of the IBM common stock (Box and Jenkins 1976, p 527) are shown in Output 8.6.1. The time series look serially uncorrelated, but the plot makes us skeptical of their independence.

With a DATA step, the stock (capital) returns are computed from the closing prices. To forecast the conditional variance, an additional 46 observations with missing values are generated.


   title 'IBM Stock Returns (daily)';
   title2 '29jun1959 - 30jun1960';
   
   data ibm;
      infile datalines eof=last;
      input x @@;
      r = dif( log( x ) );
      time = _n_-1;
      output;
      return;
   last:
      do i = 1 to 46;
         r = .;
         time + 1;
         output;
      end;
      return;
   datalines;
   ;
   
   proc gplot data=ibm;
      plot r*time / vref=0;
      symbol1 i=join v=none;
   run;

Output 8.6.1: IBM Stock Returns: Daily
autex07a.gif (6312 bytes)

The simple ARCH(2) model is estimated using the AUTOREG procedure. The MODEL statement option GARCH=(Q=2) specifies the ARCH(2) model. The OUTPUT statement with the CEV= option produces the conditional variances V. The conditional variance and its forecast is calculated using parameter estimates:

h_{t} = \hat{{\omega}} + \hat{{\alpha}}_{1}
 {\epsilon}_{t-1}^2 + \hat{{\alpha}}_{2}
 {\epsilon}_{t-2}^2
E( {\epsilon}^2_{t+d}| {\Psi}_{t}) =
\hat{{\omega}} + \sum_{i=1}^2{\hat{{\alpha}}_{i}
E( {\epsilon}^2_{t+d-i}| {\Psi}_{t})}
where d>1.

   proc autoreg data=ibm maxit=50;
      model r = / noint garch=(q=2);
      output out=a cev=v;
   run;

The parameter estimates for {{\omega}, {\alpha}_{1}}, and { {\alpha}_{2}} are 0.00011, 0.04136, and 0.06976, respectively. The normality test indicates that the conditional normal distribution may not fully explain the leptokurtosis in the stock returns (Bollerslev 1987).

The ARCH model estimates are shown in Output 8.6.2, and conditional variances are also shown in Output 8.6.3.

Output 8.6.2: ARCH(2) Estimation Results

The AUTOREG Procedure

Dependent Variable r

Ordinary Least Squares Estimates
SSE 0.03214307 DFE 254
MSE 0.0001265 Root MSE 0.01125
SBC -1558.802 AIC -1558.802
Regress R-Square 0.0000 Total R-Square 0.0000
Durbin-Watson 2.1377    
NOTE: No intercept term is used. R-squares are redefined.

Algorithm converged.

GARCH Estimates
SSE 0.03214307 Observations 254
MSE 0.0001265 Uncond Var 0.00012632
Log Likelihood 781.017441 Total R-Square 0.0000
SBC -1545.4229 AIC -1556.0349
Normality Test 105.8557 Pr > ChiSq <.0001
NOTE: No intercept term is used. R-squares are redefined.

Variable DF Estimate Standard Error t Value Approx
Pr > |t|
ARCH0 1 0.000112 7.5608E-6 14.85 <.0001
ARCH1 1 0.0413 0.0511 0.81 0.4181
ARCH2 1 0.0697 0.0432 1.62 0.1062

Output 8.6.3: Conditional Variance for IBM Stock Prices
autex07c.gif (5261 bytes)

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