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**Modelling the persistence of conditional variances.**
*(English)*
Zbl 0619.62105

The authors survey the recent stand in ARCH-type modelling and they also present new developments of this area of research. The increasing interest in ARCH-type models and its extensions stems from microeconomic theory in which risk-averse agents have to make decisions based upon the distribution of a random variable some time in the future. This requires the adequate modelling of risk and uncertainty.

The authors start with a simple version of the ARCH and generalized ARCH (GARCH) formulations for modelling conditional variances. The properties of the models are discussed and by use of asset pricing theory an example is given. The next part of the paper defines a new class of models which are defined as integrated in variance. These include as a special case the variance analogue of a unit root in the mean.

A further extension consists of generalizing the conditional density from a normal to a Student-\(t\) distribution with unknown degrees of freedom. Procedures for estimating these unknown degrees of freedom are given. With these estimates implications about the conditional kurtosis of these models and time aggregated models can be drawn. A final generalization allows the conditional heteroscedasticity to be a nonlinear function of the squared innovations.

The paper is discussed by F. X. Diebold, John Geweke, David F. Hendry, S. G. Pantula and St. E. Zin.

The authors start with a simple version of the ARCH and generalized ARCH (GARCH) formulations for modelling conditional variances. The properties of the models are discussed and by use of asset pricing theory an example is given. The next part of the paper defines a new class of models which are defined as integrated in variance. These include as a special case the variance analogue of a unit root in the mean.

A further extension consists of generalizing the conditional density from a normal to a Student-\(t\) distribution with unknown degrees of freedom. Procedures for estimating these unknown degrees of freedom are given. With these estimates implications about the conditional kurtosis of these models and time aggregated models can be drawn. A final generalization allows the conditional heteroscedasticity to be a nonlinear function of the squared innovations.

The paper is discussed by F. X. Diebold, John Geweke, David F. Hendry, S. G. Pantula and St. E. Zin.

Reviewer: H.S.Buscher

### MSC:

62P20 | Applications of statistics to economics |

91B24 | Microeconomic theory (price theory and economic markets) |

### Keywords:

autoregressive conditional heteroscedasticity; nonlinear conditional heteroscedasticity; exchange rate determination; ARCH-type models; modelling of risk and uncertainty; GARCH; modelling conditional variances; asset pricing theory; integrated in variance; generalizing the conditional density; Student-\(t\) distribution with unknown degrees of freedom; conditional kurtosis; time aggregated models
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\textit{R. F. Engle} and \textit{T. Bollerslev}, Econom. Rev. 5, 1--50 (1986; Zbl 0619.62105)

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