Modelling the persistence of conditional variances

TLDR

The discussion is motivated by a simple asset‑pricing theory particularly suited to futures contracts with risk‑averse agents. The paper reviews current research on building conditional variance models using ARCH and GARCH frameworks. The authors introduce a new class of integrated variance models that extend ARCH/GARCH by incorporating a variance unit‑root analogue, Student‑t conditional densities, and nonlinear functions of squared innovations. The models are shown to be theoretically significant for asset pricing and empirically relevant, with estimated degrees of freedom revealing conditional kurtosis patterns and empirical estimates on the USD/CHF exchange rate demonstrating their practical applicability.

Abstract

This paper will discuss the current research in building models of conditional variances using the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) formulations. The discussion will be motivated by a simple asset pricing theory which is particularly appropriate for examining futures contracts with risk averse agents. A new class of models defined to be integrated in variance is then introduced. This new class of models includes the variance analogue of a unit root in the mean as a special case. The models are argued to be both theoretically important for the asset pricing models and empirically relevant. The conditional density is then generalized from a normal to a Student-t with unknown degrees of freedom. By estimating the degrees of freedom, implications about the conditional kurtosis of these models and time aggregated models can be drawn. A further generalization allows the conditional variance to be a non-linear function of the squared innovations. Throughout empirical e imates of the logarithm of the exchange rate between the U.S. dollar and the Swiss franc are presented to illustrate the models.