We will specify our mean equation with a simple constant:

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:

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-distribution suggests that the distribution of the standardized errors departs significantly from normality.

To test whether there any remaining ARCH effects in the residuals, select View/Residual Diagnostics/ARCH LM Test... and specify the order to test. EViews will open the general Heteroskedasticity Tests dialog opened to the ARCH page. Enter “7” in the dialog for the number of lags and click on OK.

The top portion of the output from testing up-to an ARCH(7) is given by:

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/Graph... and Quantile-Quantile/Theoretical from the list of graph types on the left-hand side of the dialog.

If the residuals are normally distributed, the points in the QQ-plots should lie alongside a straight line; see “Quantile-Quantile (Theoretical)” for details on QQ-plots. The plot indicates that it is primarily large negative shocks that are driving the departure from normality. Note that we have modified the QQ-plot slightly by setting identical axes to facilitate comparison with the diagonal line.

We can also plot the residuals against the quantiles of the t-distribution. Instead of using the built-in QQ-plot for the t-distribution, you could instead simulate a draw from a t-distribution and examine whether the quantiles of the simulated observations match the quantiles of the residuals (this technique is useful for distributions not supported by EViews). The command:

simulates a random draw from the t-distribution with 6.7 degrees of freedom. Then, create a group containing the series RESID02 and TDIST. Select View/Graph... and choose Quantile-Quantile from the left-hand side of the dialog and Empirical from the Q-Q graph dropdown on the right-hand side.

The large negative residuals more closely follow a straight line. On the other hand, one can see a slight deviation from t-distribution for large positive shocks. This is expected, as the previous QQ-plot suggested that, with the exception of the large negative shocks, the residuals were close to normally distributed.

To see how the model might fit real data, we examine static forecasts for out-of-sample data. Click on the Forecast button on the equation toolbar, type in “SPX_VOL” in the GARCH field to save the forecasted 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 for the years 2000 and 2001 provides an indication of the model’s ability to track variations in market volatility.

The following commands will (as of 2004-04-03) contact the Federal Reserve Data server, fetch the daily SP500 index data (SP500) and the monthly industrial production data (INDPRO) and create new returns series based on the logarithmic changes in those series:

We may then estimate a MIDAS GARCH model for the daily returns, using 12 lags of the monthly industrial production changes as explanatory variables in the permanent component specification.

The MIDAS GARCH dialog may be filled out as follows, with D_INDPRO on the MONTHLY page specified as our low-frequency MIDAS regressor:

and click on OK to perform the estimation using the remaining default settings.

Alternately, you may issue the following commands to perform the estimation procedure and display results:

Following estimation, EViews will display the results view. The top portion of the results shows the estimation settings, sample information, computation information, and a brief text description of the transitory and permanent component specifications:

The next section shows standard coefficient results for the mean equation, transitory portion of the variance equation, and the permanent component, as well as summary statistics for the estimated equation:

As is the case with other EViews estimators, you may find it useful to use the Coefficient Labels view to see the mapping of the results in the output to the underlying coefficients.