610—Chapter 20. ARCH and GARCH Estimation

In this example, the sum of the ARCH and GARCH coefficients (α + β ) is very close to one, indicating that volatility shocks are quite persistent. This result is often observed in high frequency financial data.

Working with ARCH Models

Once your model has been estimated, EViews provides a variety of views and procedures for inference and diagnostic checking.

Views of ARCH Models

•Actual, Fitted, Residual view displays the residuals in various forms, such as table, graphs, and standardized residuals. You can save the residuals as a named series in your workfile using a procedure (see below).

•GARCH Graph plots the one-step ahead standard deviation σt or variance σ2t for each observation in the sample. The observation at period t is the forecast for t made using information available in t − 1 . You can save the conditional standard

deviations or variances as named series in your workfile using a procedure (see below). If the specification is for a component model, EViews will also display the permanent and transitory components.

•Covariance Matrix displays the estimated coefficient covariance matrix. Most ARCH models (except ARCH-M models) are block diagonal so that the covariance between the mean coefficients and the variance coefficients is very close to zero. If you include a constant in the mean equation, there will be two C’s in the covariance matrix; the first C is the constant of the mean equation, and the second C is the constant of the variance equation.

•Coefficient Tests carries out standard hypothesis tests on the estimated coefficients. See “Coefficient Tests” on page 570 for details. Note that the likelihood ratio tests are not appropriate under a quasi-maximum likelihood interpretation of your results.

•Residual Tests/Correlogram–Q-statistics displays the correlogram (autocorrelations and partial autocorrelations) of the standardized residuals. This view can be used to test for remaining serial correlation in the mean equation and to check the specification of the mean equation. If the mean equation is correctly specified, all Q-statistics should not be significant. See “Correlogram” on page 326 for an explanation of correlograms and Q-statistics.

•Residual Tests/Correlogram Squared Residuals displays the correlogram (autocorrelations and partial autocorrelations) of the squared standardized residuals. This view can be used to test for remaining ARCH in the variance equation and to check the specification of the variance equation. If the variance equation is correctly specified, all Q-statistics should not be significant. See “Correlogram” on page 326 for an

612—Chapter 20. ARCH and GARCH Estimation

these forecasts in your workfile, type a name in the corresponding dialog box. If you choose the Do graph option, EViews displays the graphs of the forecasts and two standard deviation bands for the mean forecast.

Note that the squared residuals 2t may not be available for presample values or when computing dynamic forecasts. In such cases, EViews will replaced the term by its expect value. In the simple GARCH(p, q) case, for example, the expected value of the squared residual is the fitted variance, e.g., E( 2t ) = σ2t . In other models, the expected value of the residual term will differ depending on the distribution and, in some cases, the estimated parameters of the model.

For example, to construct dynamic forecasts of SPX using the previously estimated model, click on Forecast and fill in the Forecast dialog, setting the sample after the estimation period. If you choose Do graph, the equation view changes to display the forecast results. Here, we compute the forecasts from Jan. 1, 2000 to Jan. 1, 2001, and display them side-by-side.

The first graph is the forecast of SPX (SPXF) from the mean equation with two stan-

dard deviation bands. The second graph is the forecast of the conditional variance

σ2t .

Additional ARCH Models

In addition to the standard GARCH specification, EViews has the flexibility to estimate several other variance models. These include TARCH, EGARCH, PARCH, and component GARCH. For each of these models, the user has the ability to choose the order, if any, of asymmetry.

Additional ARCH Models—613

The Threshold GARCH (TARCH) Model

TARCH or Threshold ARCH and Threshold GARCH were introduced independently by Zakoïan (1994) and Glosten, Jaganathan, and Runkle (1993). The generalized specification for the conditional variance is given by:

where I–t = 1 if t < 0 and 0 otherwise.

In this model, good news, t − i > 0 , and bad news. t − i < 0 , have differential effects on the conditional variance; good news has an impact of αi , while bad news has an impact of αi + γi . If γi > 0 , bad news increases volatility, and we say that there is a leverage effect for the i-th order. If γi ≠ 0 , the news impact is asymmetric.

Note that GARCH is a special case of the TARCH model where the threshold term is set to zero. To estimate a TARCH model, specify your GARCH model with ARCH and GARCH order and then change the Threshold order to the desired value.

The Exponential GARCH (EGARCH) Model

The EGARCH or Exponential GARCH model was proposed by Nelson (1991). The specification for the conditional variance is:

Note that the left-hand side is the log of the conditional variance. This implies that the leverage effect is exponential, rather than quadratic, and that forecasts of the conditional variance are guaranteed to be nonnegative. The presence of leverage effects can be tested by the hypothesis that γi < 0 . The impact is asymmetric if γi ≠ 0 .

There are a couple of differences between the EViews specification of the EGARCH model and the original Nelson model. First, Nelson assumes that the t follows a Generalized Error Distribution (GED), while EViews gives you a choice of normal, Student’s t-distribu- tion, or GED. Second, Nelson's specification for the log conditional variance is a restricted version of:

which differs slightly from the specification above. Estimating this model will yield identical estimates to those reported by EViews except for the intercept term w , which will dif-

614—Chapter 20. ARCH and GARCH Estimation

fer in a manner that depends upon the distributional assumption and the order p . For example, in a p = 1 model with a normal distribution, the difference will be α12 ⁄ π .

To estimate an EGARCH model, simply select the EGARCH in the model specification combo box and enter the orders for the ARCH, GARCH and the Asymmetry order.

The Power ARCH (PARCH) Model

Taylor (1986) and Schwert (1989) introduced the standard deviation GARCH model, where the standard deviation is modeled rather than the variance. This model, along with several other models, is generalized in Ding et al. (1993) with the Power ARCH specification. In the Power ARCH model, the power parameter δ of the standard deviation can be estimated rather than imposed, and the optional γ parameters are added to capture asymmetry of up to order r :

the PARCH model is simply a standard GARCH specification. As in the previous models, the asymmetric effects are present if γ ≠ 0 .

To estimate this model, simply select the PARCH in the model specification combo box and input the orders for the ARCH, GARCH and Asymmetric terms. EViews provides you with the option of either estimating or fixing a value for δ . To estimate the Taylor-Schwert's

Additional ARCH Models—615

model, for example, you will to set the order of the asymmetric terms to zero and will set δ to 1.

The Component GARCH (CGARCH) Model

The conditional variance in the GARCH(1, 1) model:

shows mean reversion to ω , which is a constant for all time. By contrast, the component model allows mean reversion to a varying level mt , modeled as:

σ2t − mt = ω + α( 2t − 1 − ω ) + β( σ2t − 1 − ω)

(20.22)

mt = ω + ρ( mt − 1 − ω ) + φ( 2t − 1− σ2t − 1) .

Here σ2t is still the volatility, while qt takes the place of ω and is the time varying longrun volatility. The first equation describes the transitory component, σ2t − qt , which converges to zero with powers of (α + β ). The second equation describes the long run component mt , which converges to ω with powers of ρ . ρ is typically between 0.99 and 1 so that mt approaches ω very slowly. We can combine the transitory and permanent equations and write:

which shows that the component model is a (nonlinear) restricted GARCH(2, 2) model.

616—Chapter 20. ARCH and GARCH Estimation

To select the Component ARCH model, simply choose Component ARCH(1,1) in the Model combo box. You can include exogenous variables in the conditional variance equation of component models, either in the permanent or transitory equation (or both). The variables in the transitory equation will have an impact on the short run movements in volatility, while the variables in the permanent equation will affect the long run levels of volatility.

An asymmetric Component ARCH model may be estimated by checking the Include threshold term checkbox. This option combines the component model with the asymmetric TARCH model, introducing asymmetric effects in the transitory equation and estimates models of the form:

where z are the exogenous variables and d is the dummy variable indicating negative shocks. γ > 0 indicates the presence of transitory leverage effects in the conditional variance.

User Specified Models

In some cases, you might wish to estimate an ARCH model not mentioned above, for example a special variant of PARCH. Many other ARCH models can be estimated using the logl object. For example, Chapter 22, “The Log Likelihood (LogL) Object”, beginning on page 671 contains examples of using logl objects for simple bivariate GARCH models.