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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

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:

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 , and 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 ApproxPr > |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

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