Additional ARCH Models

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

The Integrated GARCH (IGARCH) Model

If one restricts the parameters of the GARCH model to sum to one and drop the constant term

(25.19) |

such that

(25.20) |

then we have an integrated GARCH. This model was originally described in Engle and Bollerslev (1986). To estimate this model, select IGARCH in the Restrictions drop-down menu for the GARCH/TARCH model.

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:

(25.21) |

where if and 0 otherwise.

In this model, good news, , and bad news. , have differential effects on the conditional variance; good news has an impact of , while bad news has an impact of . If , bad news increases volatility, and we say that there is a leverage effect for the i-th order. If , 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:

(25.22) |

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 . The impact is asymmetric if .

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

which is an alternative parameterization of the specification above. Estimating the latter model will yield identical estimates to those reported by EViews except for the intercept term , which will differ in a manner that depends upon the distributional assumption and the order . For example, in a model with a normal distribution, the difference will be .

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

Notice that we have specified the mean equation using an explicit expression. Using the explicit expression is for illustration purposes only; we could just as well entered “dlog(ibm) c dlog(spx)” as our specification.

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 of the standard deviation can be estimated rather than imposed, and the optional parameters are added to capture asymmetry of up to order :

(25.23) |

where , for , for all , and .

The symmetric model sets for all . Note that if and for all , the PARCH model is simply a standard GARCH specification. As in the previous models, the asymmetric effects are present if .

To estimate this model, simply select the PARCH in the model specification dropdown menu 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 . To estimate the Taylor-Schwert's model, for example, you will to set the order of the asymmetric terms to zero and will set to 1.

The Component GARCH (CGARCH) Model

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

(25.24) |

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

(25.25) |

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

(25.26) |

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

To select the Component ARCH model, simply choose Component ARCH(1,1) in the Model dropdown menu. 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:

(25.27) |

where are the exogenous variables and is the dummy variable indicating negative shocks. 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, “The Log Likelihood (LogL) Object” contains examples of using logl objects for simple bivariate GARCH models.