Examples—617

Examples

For the variance specification, we employ an EGARCH(1, 1) model:

When we previously estimated a GARCH(1,1) model with the data, the standardized residual showed evidence of excess kurtosis. To model the thick tail in the residuals, we will assume that the errors follow a Student's t-distribution.

To estimate this model, open the GARCH estimation dialog, enter the mean specification:

dlog(spx) c

select the EGARCH method, enter 1 for the ARCH and GARCH orders and the Asymmetric order, and select Student’s t for the Error distribution. Click on OK to continue.

EViews displays the results of the estimation procedure. The top portion contains a description of the estimation specification, including the estimation sample, error distribution assumption, and backcast assumption.

Below the header information are the results for the mean and the variance equations, followed by the results for any distributional parameters. Here, we see that the relatively small degrees of freedom parameter for the t-distributiont suggests that the distribution of the standardized errors departs significantly from normality.

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Method: ARCH - Student's t distribution (Marquardt)

Date: 09/16/03 Time: 13:52

Sample: 1/02/1990 12/31/1999

Included observations: 2528

Convergence achieved after 18 iterations

Variance backcast: ON

LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) +

C(4)*RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))

To test whether there any remaining ARCH effects in the residuals, select View/Residual Tests/ARCH LM Test... and specify the order to test. The top portion of the output from testing up-to an ARCH(7) is given by:

ARCH Test:

so there is little evidence of remaining ARCH effects.

One way of further examining the distribution of the residuals is to plot the quantiles. First, save the standardized residuals by clicking on Proc/Make Residual Series..., select the Standardized option, and specify a name for the resulting series. EViews will create a series containing the desired residuals; in this example, we create a series named RESID02. Then open the residual series window and select View/Distribution/Quantile-Quantile Graphs... and tell EViews the distribution whose quantiles you wish to plot against, for example, the Normal distribution.

620—Chapter 20. ARCH and GARCH Estimation

casted conditional variance, change the sample to the post-estimation sample period “1/1/ 2000 1/1/2002” and click on Static to select a static forecast.

Since the actual volatility is unobserved, we will use the squared return series (DLOG(SPX)^2) as a proxy for the realized volatility. A plot of the proxy against the forecasted volatility provides an indication of the model’s ability to track variations in market volatility.

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